{"assistant":{"explanations":[{"calculationRefs":["defined"],"text":"Три посылки: обе категории абелевы, достаточно инъективных объектов, функтор аддитивен."},{"calculationRefs":["resolution"],"text":"Достаточность инъективных объектов даёт резольвенту для каждого объекта категории."},{"calculationRefs":["r1"],"text":"Первый производный считается как когомология комплекса, полученного применением функтора к резольвенте."},{"calculationRefs":["acyclic"],"text":"На инъективном объекте производные положительной степени обращаются в нуль."},{"calculationRefs":["r0"],"text":"Совпадение нулевой степени с самим функтором — ровно свойство левой точности."},{"calculationRefs":["long-exact"],"text":"Короткая точная последовательность объектов порождает длинную точную последовательность производных."},{"calculationRefs":["negative"],"text":"Ниже нулевой степени производных функторов нет."}],"origin":"assistant","summary":"Правый производный функтор определён всюду, когда обе категории абелевы, в источнике достаточно инъективных объектов, а сам функтор аддитивен. 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category"},{"language":"ru","status":"unofficial","text":"естественное отображение \\(Coim(f) → Im(f)\\) есть изоморфизм для всех морфизмов \\(f\\) категории"}],"name":"coimage_to_image_isomorphism","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#coimage_to_image_isomorphism/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#enough_injectives","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category has enough injectives: every object \\(A\\) has an injective morphism \\(A → J\\) into an injective object \\(J\\)"},{"language":"ru","status":"unofficial","text":"в категории достаточно инъективных: у всякого объекта \\(A\\) есть инъективный морфизм \\(A → J\\) в инъективный объект \\(J\\)"}],"name":"enough_injectives","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#enough_injectives/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_cokernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all cokernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все коядра"}],"name":"has_all_cokernels","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#has_all_cokernels/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_kernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all kernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все 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0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): если \\(RF\\) определён всюду, то \\(R^iF = 0\\) при \\(i < 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RFEverywhereDefined","kind":"rule","labels":[{"language":"en","status":"official","text":"05TI (2): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is an additive functor into an abelian category, then \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05TI (2): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) — аддитивный функтор в абелеву категорию, то \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TE (1): the functors \\(R^iF\\), \\(i ≥ 0\\), carry a canonical structure of a cohomological \\(δ\\)-functor (010Q)"},{"language":"ru","status":"unofficial","text":"05TE (1): функторы \\(R^iF\\), \\(i ≥ 0\\), несут каноническую структуру когомологического \\(δ\\)-функтора (010Q)"}],"name":"derived_functors_form_delta_functor","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (1): \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): \\(R^iF = 0\\) при \\(i < 0\\)"}],"name":"negative_derived_functors_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SV: the right derived functor \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05SV: правый производный функтор \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"name":"rf_everywhere_defined","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4","urn:stacks:clir:derived-functors#ST_05TD","urn:stacks:clir:derived-functors#ST_05TE","urn:stacks:clir:derived-functors#ST_05TI"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_00ZY","urn:stacks:clir:categories#ST_0104","urn:stacks:clir:categories#ST_0109"],"jurisdiction":"none","namespace":"urn:stacks:clir:categories","package":"stacks-categories","title":"Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина"},{"contributed":false,"fragmentCount":17,"fragments":[],"jurisdiction":"none","namespace":"urn:stacks:clir:category-theory","package":"stacks-category-theory","title":"Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:7299d7d4001b7f6154f0372b741e9c7898ae828311ac9148a175f4d580767b11","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317","resultHash":"sha256:2770eae7501dfd5877e5a407d53dd2a4ca8610caff6b6000c2eb16d65aecbba3","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:f"],"facts":[{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"enough_injectives"},{"args":["urn:case:stacks:f","urn:case:stacks:cat-a","urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"functor_between"},{"args":["urn:case:stacks:f"],"package":"stacks-categories","predicate":"homomorphism_on_hom_groups"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-derived-functors","predicate":"rf_everywhere_defined","proof":true},"sources":[{"contentHash":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SV","kind":"fragment","locator":"tag/05SV","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-everywhere-defined}\nIn\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.\n\\end{definition}"}]},{"contentHash":"sha256:7b734956d4fc9c61b7d92b47efd6278d84ca8bed87f9ecb3d0b1bb74c80f8ea8","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"situation","id":"urn:stacks:clir:derived-functors#ST_05T4","kind":"fragment","locator":"tag/05T4","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6","language":"en","status":"official","text":"\\begin{situation}\n\\label{situation-classical}\nHere $F : \\mathcal{A} \\to \\mathcal{B}$ is an additive functor between\nabelian categories. This induces exact functors\n$$\nF : K(\\mathcal{A}) \\to K(\\mathcal{B}), \\quad\nK^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B}), \\quad\nK^{-}(\\mathcal{A}) \\to K^{-}(\\mathcal{B}).\n$$\nSee Lemma \\ref{lemma-additive-exact-homotopy-category}.\nWe also denote $F$ the composition $K(\\mathcal{A}) \\to D(\\mathcal{B})$,\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$, and\n$K^{-}(\\mathcal{A}) \\to D^-(\\mathcal{B})$ of $F$ with the localization\nfunctor $K(\\mathcal{B}) \\to D(\\mathcal{B})$, etc. This situation leads\nto four derived functors we will consider in the following.\n\\begin{enumerate}\n\\item The right derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The right derived functor of\n$F : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^{+}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^-(\\mathcal{A})$.\n\\end{enumerate}\nEach of these cases is an example of\nSituation \\ref{situation-derived-functor}.\n\\end{situation}"}]},{"contentHash":"sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TD","kind":"fragment","locator":"tag/05TD","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-left-exact-higher-derived}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TE","kind":"fragment","locator":"tag/05TE","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-right-derived-delta-functor}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TI","kind":"fragment","locator":"tag/05TI","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-enough-injectives-right-derived}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"}],"text":"rf_everywhere_defined: **TRUE_ONLY** — установлено\nВыведено правом: additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f)\nПрименены правила: AdditiveByFiniteProducts, AdditiveByHomGroups, abelian_category/sufficient, DerivedFormDeltaFunctor, NegativeDerivedVanish, RFEverywhereDefined\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 24 узлов — поле evaluation готово для 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category"},{"language":"ru","status":"unofficial","text":"естественное отображение \\(Coim(f) → Im(f)\\) есть изоморфизм для всех морфизмов \\(f\\) категории"}],"name":"coimage_to_image_isomorphism","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#coimage_to_image_isomorphism/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#enough_injectives","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category has enough injectives: every object \\(A\\) has an injective morphism \\(A → J\\) into an injective object \\(J\\)"},{"language":"ru","status":"unofficial","text":"в категории достаточно инъективных: у всякого объекта \\(A\\) есть инъективный морфизм \\(A → J\\) в инъективный объект \\(J\\)"}],"name":"enough_injectives","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#enough_injectives/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_cokernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all cokernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все коядра"}],"name":"has_all_cokernels","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#has_all_cokernels/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_kernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all kernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все ядра"}],"name":"has_all_kernels","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#has_all_kernels/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#preadditive_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"00ZY: every \\(Mor(x, y)\\) is an abelian group and composition is bilinear"},{"language":"ru","status":"unofficial","text":"00ZY: каждое \\(Mor(x, y)\\) — абелева группа, композиция билинейна"}],"name":"preadditive_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#preadditive_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#Category","kind":"type_decl","labels":[{"language":"en","status":"official","text":"category"},{"language":"ru","status":"unofficial","text":"категория"}],"name":"Category","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#Obj","kind":"type_decl","labels":[{"language":"en","status":"official","text":"object of a category"},{"language":"ru","status":"unofficial","text":"объект категории"}],"name":"Obj","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#has_finite_products","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"001S: products \\(x × y\\) exist for all pairs of objects, hence all finite products"},{"language":"ru","status":"unofficial","text":"001S: произведения \\(x × y\\) существуют для всех пар объектов, а значит все конечные произведения"}],"name":"has_finite_products","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#has_finite_products/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#object_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the object belongs to the category: \\(x ∈ Ob(C)\\)"},{"language":"ru","status":"unofficial","text":"объект принадлежит категории: \\(x ∈ Ob(C)\\)"}],"name":"object_of","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#object_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:category-theory#Obj"}},{"id":"urn:stacks:clir:category-theory#object_of/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#InjectiveResolutionsExist","kind":"rule","labels":[{"language":"en","status":"official","text":"013K (1): in an abelian category with enough injectives any object has an injective resolution"},{"language":"ru","status":"unofficial","text":"013K (1): в абелевой категории с достаточным запасом инъективных у всякого объекта есть инъективная резольвента"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#has_injective_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the object has an injective resolution"},{"language":"ru","status":"unofficial","text":"у объекта есть инъективная резольвента"}],"name":"has_injective_resolution","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#has_injective_resolution/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_013K"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_00ZY","urn:stacks:clir:categories#ST_0104","urn:stacks:clir:categories#ST_0109"],"jurisdiction":"none","namespace":"urn:stacks:clir:categories","package":"stacks-categories","title":"Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина"},{"contributed":false,"fragmentCount":17,"fragments":[],"jurisdiction":"none","namespace":"urn:stacks:clir:category-theory","package":"stacks-category-theory","title":"Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:16d96832d4b46160686e000373db91605fb7fe8a63485616e294fea65e774545","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317","resultHash":"sha256:f65c6402e0f1650ff812e9ad868c34d07a4a8a444637c974636851f05dfd808d","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:x"],"facts":[{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"enough_injectives"},{"args":["urn:case:stacks:x","urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"object_of"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-derived-functors","predicate":"has_injective_resolution","proof":true},"sources":[{"contentHash":"sha256:9698ad04b03a0ca888a87e8f88e9925a856764046ae30cfb569730f33de5c92d","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_013K","kind":"fragment","locator":"tag/013K","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:869cffaf0235a6f9d2d38c26d042d9ff1c0bbfae733cd1cdcc72af28b5bc8fa4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-injective-resolutions-exist}\nLet $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\n\\begin{enumerate}\n\\item Any object of $\\mathcal{A}$ has an injective resolution.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then\n$K^\\bullet$ has an injective resolution.\n\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n < a$, then\nthere exists an injective resolution $\\alpha : K^\\bullet \\to I^\\bullet$\nwith $I^n = 0$ for $n < a$ such that each $\\alpha^n : K^n \\to I^n$ is\ninjective.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"}],"text":"has_injective_resolution: **TRUE_ONLY** — установлено\nВыведено правом: additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); has_injective_resolution(urn:case:stacks:x)\nПрименены правила: AdditiveByFiniteProducts, abelian_category/sufficient, InjectiveResolutionsExist\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 12 узлов — поле evaluation готово для 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Ob(C)\\)"}],"name":"object_of","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#object_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:category-theory#Obj"}},{"id":"urn:stacks:clir:category-theory#object_of/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#Complex","kind":"type_decl","labels":[{"language":"en","status":"official","text":"cochain complex"},{"language":"ru","status":"unofficial","text":"коцепной комплекс"}],"name":"Complex","package":"urn:stacks:clir:derived-functors"},{"id":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"05TE (1): if \\(RF\\) is everywhere defined, the \\(R^iF\\) come equipped with a canonical structure of a \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"05TE (1): если \\(RF\\) определён всюду, \\(R^iF\\) снабжены канонической структурой \\(δ\\)-функтора"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#InjectiveResolutionComputesRF","kind":"rule","labels":[{"language":"en","status":"official","text":"05TH (1) with 013I: an injective resolution \\(I^•\\) of \\(A\\) is a bounded below complex of injectives, hence computes \\(RF\\) for any additive \\(F : A → B\\)"},{"language":"ru","status":"unofficial","text":"05TH (1) с 013I: инъективная резольвента \\(I^•\\) объекта \\(A\\) — ограниченный снизу комплекс инъективных, потому вычисляет \\(RF\\) для любого аддитивного \\(F : A → B\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#InjectiveResolutionResolves","kind":"rule","labels":[{"language":"en","status":"official","text":"013I: an injective resolution \\(A → I^•\\) is a quasi-isomorphism \\(A[0] → I^•\\), and \\(I^•\\) is bounded below by (1)"},{"language":"ru","status":"unofficial","text":"013I: инъективная резольвента \\(A → I^•\\) — квазиизоморфизм \\(A[0] → I^•\\), и \\(I^•\\) ограничен снизу по (1)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#InjectiveResolutionsExist","kind":"rule","labels":[{"language":"en","status":"official","text":"013K (1): in an abelian category with enough injectives any object has an injective resolution"},{"language":"ru","status":"unofficial","text":"013K (1): в абелевой категории с достаточным запасом инъективных у всякого объекта есть инъективная резольвента"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (1): if \\(RF\\) is everywhere defined, then \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): если \\(RF\\) определён всюду, то \\(R^iF = 0\\) при \\(i < 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RFEverywhereDefined","kind":"rule","labels":[{"language":"en","status":"official","text":"05TI (2): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is an additive functor into an abelian category, then \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05TI (2): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) — аддитивный функтор в абелеву категорию, то \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RightDerivedByResolution","kind":"rule","labels":[{"language":"en","status":"official","text":"015A with 05SX: if \\(K^•\\) resolves \\(A\\) and computes \\(RF\\), then \\(R^nF(A) = H^n(F(K^•))\\)"},{"language":"ru","status":"unofficial","text":"015A с 05SX: если \\(K^•\\) резольвирует \\(A\\) и вычисляет \\(RF\\), то \\(R^nF(A) = H^n(F(K^•))\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#acyclic_in_positive_degrees","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"013I (4): \\(H^i(I^•) = 0\\) for \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"013I (4): \\(H^i(I^•) = 0\\) при \\(i > 0\\)"}],"name":"acyclic_in_positive_degrees","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#acyclic_in_positive_degrees/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#applied_complex","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F(K^•)\\) is the given complex: \\(F\\) applied termwise to \\(K^•\\)"},{"language":"ru","status":"unofficial","text":"\\(F(K^•)\\) — данный комплекс: \\(F\\), применённый к \\(K^•\\) почленно"}],"name":"applied_complex","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#applied_complex/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"urn:stacks:clir:derived-functors#applied_complex/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}},{"id":"urn:stacks:clir:derived-functors#applied_complex/arg/fk","labels":[],"name":"fk","type":{"name":"urn:stacks:clir:derived-functors#Complex"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#augmentation_onto_kernel","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"013I (3): the map \\(A → I^0\\) is an isomorphism onto \\(Ker(d^0)\\)"},{"language":"ru","status":"unofficial","text":"013I (3): отображение \\(A → I^0\\) — изоморфизм на \\(Ker(d^0)\\)"}],"name":"augmentation_onto_kernel","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#augmentation_onto_kernel/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}},{"id":"urn:stacks:clir:derived-functors#augmentation_onto_kernel/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#cohomology_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(H^n(K^•)\\) is the given object"},{"language":"ru","status":"unofficial","text":"\\(H^n(K^•)\\) — данный объект"}],"name":"cohomology_of","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#cohomology_of/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}},{"id":"urn:stacks:clir:derived-functors#cohomology_of/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}},{"id":"urn:stacks:clir:derived-functors#cohomology_of/arg/h","labels":[],"name":"h","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#complex_in","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(K^•\\) is a complex in the category"},{"language":"ru","status":"unofficial","text":"\\(K^•\\) — комплекс в категории"}],"name":"complex_in","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#complex_in/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}},{"id":"urn:stacks:clir:derived-functors#complex_in/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:derived-functors#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#computes_rf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SX: the complex \\(K^•\\) computes \\(RF\\): \\(RF\\) is defined at \\(K^•\\) and \\(F(K^•) → RF(K^•)\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05SX: комплекс \\(K^•\\) вычисляет \\(RF\\): \\(RF\\) определён в \\(K^•\\) и \\(F(K^•) → RF(K^•)\\) — изоморфизм"}],"name":"computes_rf","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#computes_rf/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}},{"id":"urn:stacks:clir:derived-functors#computes_rf/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TE (1): the functors \\(R^iF\\), \\(i ≥ 0\\), carry a canonical structure of a cohomological \\(δ\\)-functor (010Q)"},{"language":"ru","status":"unofficial","text":"05TE (1): функторы \\(R^iF\\), \\(i ≥ 0\\), несут каноническую структуру когомологического \\(δ\\)-функтора (010Q)"}],"name":"derived_functors_form_delta_functor","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#has_injective_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the object has an injective resolution"},{"language":"ru","status":"unofficial","text":"у объекта есть инъективная резольвента"}],"name":"has_injective_resolution","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#has_injective_resolution/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#injective_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"013I: an injective resolution of \\(A\\) is a complex \\(I^•\\) with \\(A → I^0\\) such that (1) \\(I^n = 0\\) for \\(n < 0\\), (2) each \\(I^n\\) is injective, (3) \\(A → I^0\\) is an isomorphism onto \\(Ker(d^0)\\), (4) \\(H^i(I^•) = 0\\) for \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"013I: инъективная резольвента \\(A\\) — комплекс \\(I^•\\) с \\(A → I^0\\), такой что (1) \\(I^n = 0\\) при \\(n < 0\\), (2) каждый \\(I^n\\) инъективен, (3) \\(A → I^0\\) — изоморфизм на \\(Ker(d^0)\\), (4) \\(H^i(I^•) = 0\\) при \\(i > 0\\)"}],"name":"injective_resolution","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#injective_resolution/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}},{"id":"urn:stacks:clir:derived-functors#injective_resolution/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#injective_resolution/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"013I: an injective resolution of \\(A\\) is a complex \\(I^•\\) with \\(A → I^0\\) such that (1) \\(I^n = 0\\) for \\(n < 0\\), (2) each \\(I^n\\) is injective, (3) \\(A → I^0\\) is an isomorphism onto \\(Ker(d^0)\\), (4) \\(H^i(I^•) = 0\\) for \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"013I: инъективная резольвента \\(A\\) — комплекс \\(I^•\\) с \\(A → I^0\\), такой что (1) \\(I^n = 0\\) при \\(n < 0\\), (2) каждый \\(I^n\\) инъективен, (3) \\(A → I^0\\) — изоморфизм на \\(Ker(d^0)\\), (4) \\(H^i(I^•) = 0\\) при \\(i > 0\\)"}],"package":"urn:stacks:clir:derived-functors"},{"id":"urn:stacks:clir:derived-functors#injective_resolution/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"013I: an injective resolution of \\(A\\) is a complex \\(I^•\\) with \\(A → I^0\\) such that (1) \\(I^n = 0\\) for \\(n < 0\\), (2) each \\(I^n\\) is injective, (3) \\(A → I^0\\) is an isomorphism onto \\(Ker(d^0)\\), (4) \\(H^i(I^•) = 0\\) for \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"013I: инъективная резольвента \\(A\\) — комплекс \\(I^•\\) с \\(A → I^0\\), такой что (1) \\(I^n = 0\\) при \\(n < 0\\), (2) каждый \\(I^n\\) инъективен, (3) \\(A → I^0\\) — изоморфизм на \\(Ker(d^0)\\), (4) \\(H^i(I^•) = 0\\) при \\(i > 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (1): \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): \\(R^iF = 0\\) при \\(i < 0\\)"}],"name":"negative_derived_functors_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#resolution_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(K^•\\) resolves \\(A\\): \\(K^•\\) is bounded below and \\(A[0] → K^•\\) is a quasi-isomorphism"},{"language":"ru","status":"unofficial","text":"\\(K^•\\) резольвирует \\(A\\): \\(K^•\\) ограничен снизу и \\(A[0] → K^•\\) — квазиизоморфизм"}],"name":"resolution_of","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#resolution_of/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}},{"id":"urn:stacks:clir:derived-functors#resolution_of/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SV: the right derived functor \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05SV: правый производный функтор \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"name":"rf_everywhere_defined","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#right_derived","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"015A: \\(R^nF(A) = H^n(RF(A[0]))\\) is the given object"},{"language":"ru","status":"unofficial","text":"015A: \\(R^nF(A) = H^n(RF(A[0]))\\) — данный объект"}],"name":"right_derived","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#right_derived/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"urn:stacks:clir:derived-functors#right_derived/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}},{"id":"urn:stacks:clir:derived-functors#right_derived/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}},{"id":"urn:stacks:clir:derived-functors#right_derived/arg/h","labels":[],"name":"h","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#terms_injective","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"013I (2): each \\(I^n\\) is an injective object of the category"},{"language":"ru","status":"unofficial","text":"013I (2): каждый \\(I^n\\) — инъективный объект категории"}],"name":"terms_injective","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#terms_injective/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#vanishes_in_negative_degrees","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"013I (1): \\(I^n = 0\\) for \\(n < 0\\)"},{"language":"ru","status":"unofficial","text":"013I (1): \\(I^n = 0\\) при \\(n < 0\\)"}],"name":"vanishes_in_negative_degrees","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#vanishes_in_negative_degrees/arg/k","labels":[],"name":"k","type":{"name":"urn:stacks:clir:derived-functors#Complex"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_013I","urn:stacks:clir:derived-functors#ST_013K","urn:stacks:clir:derived-functors#ST_015A","urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05SX","urn:stacks:clir:derived-functors#ST_05T4","urn:stacks:clir:derived-functors#ST_05TD","urn:stacks:clir:derived-functors#ST_05TE","urn:stacks:clir:derived-functors#ST_05TH","urn:stacks:clir:derived-functors#ST_05TI"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные функторы по The Stacks Project: определения и леммы как словарь с 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государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317","resultHash":"sha256:9a35ac0bc226bc93d7a84a40c17a9ee6d2997c40b69b0e950c6ebc3437a14391","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:f",1,"urn:case:stacks:x","urn:case:stacks:h1"],"facts":[{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"enough_injectives"},{"args":["urn:case:stacks:f","urn:case:stacks:cat-a","urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"functor_between"},{"args":["urn:case:stacks:f"],"package":"stacks-categories","predicate":"homomorphism_on_hom_groups"},{"args":["urn:case:stacks:x","urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"object_of"},{"args":["urn:case:stacks:i","urn:case:stacks:cat-a"],"predicate":"complex_in"},{"args":["urn:case:stacks:i"],"predicate":"vanishes_in_negative_degrees"},{"args":["urn:case:stacks:i"],"predicate":"terms_injective"},{"args":["urn:case:stacks:x","urn:case:stacks:i"],"predicate":"augmentation_onto_kernel"},{"args":["urn:case:stacks:i"],"predicate":"acyclic_in_positive_degrees"},{"args":["urn:case:stacks:f","urn:case:stacks:i","urn:case:stacks:f-i"],"predicate":"applied_complex"},{"args":["urn:case:stacks:f-i",1,"urn:case:stacks:h1"],"predicate":"cohomology_of"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-derived-functors","predicate":"right_derived","proof":true},"sources":[{"contentHash":"sha256:d7c48bc941098921fd33924bd407abc5291d88b9236a485cdaaf7aa25e9d0832","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_013I","kind":"fragment","locator":"tag/013I","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:7330949afd1629e09a53e8833569efe91f5d8ca6b7368e0a68a87d0caa530934","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-injective-resolution}\nLet $\\mathcal{A}$ be an abelian category.\nLet $A \\in \\Ob(\\mathcal{A})$.\nAn {\\it injective resolution of $A$} is a complex\n$I^\\bullet$ together with a map $A \\to I^0$ such\nthat:\n\\begin{enumerate}\n\\item We have $I^n = 0$ for $n < 0$.\n\\item Each $I^n$ is an injective object of $\\mathcal{A}$.\n\\item The map $A \\to I^0$ is an isomorphism onto $\\Ker(d^0)$.\n\\item We have $H^i(I^\\bullet) = 0$ for $i > 0$.\n\\end{enumerate}\nHence $A[0] \\to I^\\bullet$ is a quasi-isomorphism.\nIn other words the complex\n$$\n\\ldots \\to 0 \\to A \\to I^0 \\to I^1 \\to \\ldots\n$$\nis acyclic.\nLet $K^\\bullet$ be a complex in $\\mathcal{A}$.\nAn {\\it injective resolution of $K^\\bullet$} is a complex\n$I^\\bullet$ together with a map $\\alpha : K^\\bullet \\to I^\\bullet$\nof complexes such that\n\\begin{enumerate}\n\\item We have $I^n = 0$ for $n \\ll 0$, i.e., $I^\\bullet$ is bounded below.\n\\item Each $I^n$ is an injective object of $\\mathcal{A}$.\n\\item The map $\\alpha : K^\\bullet \\to I^\\bullet$ is a\nquasi-isomorphism.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:9698ad04b03a0ca888a87e8f88e9925a856764046ae30cfb569730f33de5c92d","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_013K","kind":"fragment","locator":"tag/013K","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:869cffaf0235a6f9d2d38c26d042d9ff1c0bbfae733cd1cdcc72af28b5bc8fa4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-injective-resolutions-exist}\nLet $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\n\\begin{enumerate}\n\\item Any object of $\\mathcal{A}$ has an injective resolution.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then\n$K^\\bullet$ has an injective resolution.\n\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n < a$, then\nthere exists an injective resolution $\\alpha : K^\\bullet \\to I^\\bullet$\nwith $I^n = 0$ for $n < a$ such that each $\\alpha^n : K^n \\to I^n$ is\ninjective.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:713c25b819cf11bf73664db0382e1e6093ee2fc2aef860be0a5c1844def10500","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_015A","kind":"fragment","locator":"tag/015A","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:7ecd153a29139bad7c85e08acd4818c3aedb736903c0a95c5400227dea7d508d","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-higher-derived-functors}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories. Assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $i \\in \\mathbf{Z}$.\nThe {\\it $i$th right derived functor $R^iF$ of $F$} is the functor\n$$\nR^iF = H^i \\circ RF :\n\\mathcal{A}\n\\longrightarrow\n\\mathcal{B}\n$$\n\\end{definition}"}]},{"contentHash":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SV","kind":"fragment","locator":"tag/05SV","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-everywhere-defined}\nIn\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.\n\\end{definition}"}]},{"contentHash":"sha256:6d9640352974ec06f445076630ee0495e5463224a1f554aea25ab8abb8f231b0","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SX","kind":"fragment","locator":"tag/05SX","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:a12a542d90fce40869d878a5e1f1be4adaed0407a78b0a5d0febeb73e30ca0b5","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-computes}\nIn\nSituation \\ref{situation-derived-functor}.\n\\begin{enumerate}\n\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $RF$ if $RF$ is defined\nat $X$ and the canonical map $F(X) \\to RF(X)$ is an isomorphism.\n\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $LF$ if $LF$ is defined\nat $X$ and the canonical map $LF(X) \\to F(X)$ is an isomorphism.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:7b734956d4fc9c61b7d92b47efd6278d84ca8bed87f9ecb3d0b1bb74c80f8ea8","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"situation","id":"urn:stacks:clir:derived-functors#ST_05T4","kind":"fragment","locator":"tag/05T4","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6","language":"en","status":"official","text":"\\begin{situation}\n\\label{situation-classical}\nHere $F : \\mathcal{A} \\to \\mathcal{B}$ is an additive functor between\nabelian categories. This induces exact functors\n$$\nF : K(\\mathcal{A}) \\to K(\\mathcal{B}), \\quad\nK^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B}), \\quad\nK^{-}(\\mathcal{A}) \\to K^{-}(\\mathcal{B}).\n$$\nSee Lemma \\ref{lemma-additive-exact-homotopy-category}.\nWe also denote $F$ the composition $K(\\mathcal{A}) \\to D(\\mathcal{B})$,\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$, and\n$K^{-}(\\mathcal{A}) \\to D^-(\\mathcal{B})$ of $F$ with the localization\nfunctor $K(\\mathcal{B}) \\to D(\\mathcal{B})$, etc. This situation leads\nto four derived functors we will consider in the following.\n\\begin{enumerate}\n\\item The right derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The right derived functor of\n$F : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^{+}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^-(\\mathcal{A})$.\n\\end{enumerate}\nEach of these cases is an example of\nSituation \\ref{situation-derived-functor}.\n\\end{situation}"}]},{"contentHash":"sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TD","kind":"fragment","locator":"tag/05TD","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-left-exact-higher-derived}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TE","kind":"fragment","locator":"tag/05TE","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-right-derived-delta-functor}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:67d5bab826525b8b40aa6d952b68667e49a96a051a298a45f2890a4bfb11950e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TH","kind":"fragment","locator":"tag/05TH","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e032a20e4fb0e18232eca1300982b8c867068569810bf8e4474e48073cab1465","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-injective-acyclic}\nLet $\\mathcal{A}$ be an abelian category.\nLet $I \\in \\Ob(\\mathcal{A})$ be an injective object.\nLet $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$\nfor any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto any triangulated category $\\mathcal{D}$.\n\\item $I$ is right acyclic for any additive functor\n$F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TI","kind":"fragment","locator":"tag/05TI","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-enough-injectives-right-derived}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"}],"text":"right_derived: **TRUE_ONLY** — установлено\nВыведено правом: additive_functor(urn:case:stacks:f); injective_resolution(urn:case:stacks:i, urn:case:stacks:x); computes_rf(urn:case:stacks:i, urn:case:stacks:f); resolution_of(urn:case:stacks:i, urn:case:stacks:x); right_derived(urn:case:stacks:f, 1, urn:case:stacks:x, urn:case:stacks:h1); has_injective_resolution(urn:case:stacks:x); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f)\n…и ещё 4 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: AdditiveByFiniteProducts, AdditiveByHomGroups, abelian_category/sufficient, DerivedFormDeltaFunctor, InjectiveResolutionComputesRF, InjectiveResolutionResolves, InjectiveResolutionsExist, NegativeDerivedVanish, RFEverywhereDefined, RightDerivedByResolution, injective_resolution/sufficient\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 38 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["left_exact_functor(urn:case:stacks:f)","additive_functor(urn:case:stacks:f)","injective_object(urn:case:stacks:j)","has_injective_resolution(urn:case:stacks:j)","derived_functors_universal(urn:case:stacks:f)","right_acyclic_for(urn:case:stacks:j, urn:case:stacks:f)","rf_everywhere_defined(urn:case:stacks:f)","derived_functors_form_delta_functor(urn:case:stacks:f)","higher_derived_vanish(urn:case:stacks:f, 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\\(J\\)"}],"name":"enough_injectives","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#enough_injectives/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_cokernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all cokernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все коядра"}],"name":"has_all_cokernels","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#has_all_cokernels/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_kernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all kernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все ядра"}],"name":"has_all_kernels","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#has_all_kernels/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#injective_object","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the object is injective"},{"language":"ru","status":"unofficial","text":"объект инъективен"}],"name":"injective_object","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#injective_object/arg/j","labels":[],"name":"j","type":{"name":"urn:stacks:clir:categories#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#lifting_property","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"for every injection \\(A ↪ B\\) and every morphism \\(A → J\\) there exists a morphism \\(B → J\\) making the diagram commute"},{"language":"ru","status":"unofficial","text":"для всякой инъекции \\(A ↪ B\\) и всякого морфизма \\(A → J\\) существует морфизм \\(B → J\\), замыкающий диаграмму"}],"name":"lifting_property","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#lifting_property/arg/j","labels":[],"name":"j","type":{"name":"urn:stacks:clir:categories#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#preadditive_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"00ZY: every \\(Mor(x, y)\\) is an abelian group and composition is bilinear"},{"language":"ru","status":"unofficial","text":"00ZY: каждое \\(Mor(x, y)\\) — абелева группа, композиция билинейна"}],"name":"preadditive_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#preadditive_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#preserves_left_exactness","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"for every short exact sequence \\(0 → A → B → C → 0\\) the sequence \\(0 → F(A) → F(B) → F(C)\\) is exact"},{"language":"ru","status":"unofficial","text":"для всякой короткой точной последовательности \\(0 → A → B → C → 0\\) последовательность \\(0 → F(A) → F(B) → F(C)\\) точна"}],"name":"preserves_left_exactness","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#preserves_left_exactness/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:categories#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#Category","kind":"type_decl","labels":[{"language":"en","status":"official","text":"category"},{"language":"ru","status":"unofficial","text":"категория"}],"name":"Category","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#Functor","kind":"type_decl","labels":[{"language":"en","status":"official","text":"functor"},{"language":"ru","status":"unofficial","text":"функтор"}],"name":"Functor","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#Obj","kind":"type_decl","labels":[{"language":"en","status":"official","text":"object of a category"},{"language":"ru","status":"unofficial","text":"объект категории"}],"name":"Obj","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#functor_between","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor goes from the first category to the second: \\(F : A → B\\)"},{"language":"ru","status":"unofficial","text":"функтор действует из первой категории во вторую: \\(F : A → B\\)"}],"name":"functor_between","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#functor_between/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Functor"}},{"id":"urn:stacks:clir:category-theory#functor_between/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:category-theory#Category"}},{"id":"urn:stacks:clir:category-theory#functor_between/arg/b","labels":[],"name":"b","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#has_finite_products","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"001S: products \\(x × y\\) exist for all pairs of objects, hence all finite products"},{"language":"ru","status":"unofficial","text":"001S: произведения \\(x × y\\) существуют для всех пар объектов, а значит все конечные произведения"}],"name":"has_finite_products","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#has_finite_products/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#left_exact_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0034 (1): the functor is left exact"},{"language":"ru","status":"unofficial","text":"0034 (1): функтор точен слева"}],"name":"left_exact_functor","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#left_exact_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#object_of","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the object belongs to the category: \\(x ∈ Ob(C)\\)"},{"language":"ru","status":"unofficial","text":"объект принадлежит категории: \\(x ∈ Ob(C)\\)"}],"name":"object_of","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#object_of/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:category-theory#Obj"}},{"id":"urn:stacks:clir:category-theory#object_of/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"05TE (1): if \\(RF\\) is everywhere defined, the \\(R^iF\\) come equipped with a canonical structure of a \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"05TE (1): если \\(RF\\) определён всюду, \\(R^iF\\) снабжены канонической структурой \\(δ\\)-функтора"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"015B (4): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is left exact, then \\((R^iF, δ)\\) is a universal \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"015B (4): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) точен слева, то \\((R^iF, δ)\\) — универсальный \\(δ\\)-функтор"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic","kind":"rule","labels":[{"language":"en","status":"official","text":"015C (2): if \\(F\\) is left exact, \\(RF\\) is everywhere defined and \\(A\\) is right acyclic for \\(F\\), then \\(R^iF(A) = 0\\) for all \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"015C (2): если \\(F\\) точен слева, \\(RF\\) определён всюду и \\(A\\) правый ацикличный для \\(F\\), то \\(R^iF(A) = 0\\) при всех \\(i > 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#InjectiveIsRightAcyclic","kind":"rule","labels":[{"language":"en","status":"official","text":"05TH (2): an injective object \\(I\\) is right acyclic for any additive functor \\(F : A → B\\) into an abelian category"},{"language":"ru","status":"unofficial","text":"05TH (2): инъективный объект \\(I\\) правый ацикличный для любого аддитивного функтора \\(F : A → B\\) в абелеву категорию"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#InjectiveResolutionsExist","kind":"rule","labels":[{"language":"en","status":"official","text":"013K (1): in an abelian category with enough injectives any object has an injective resolution"},{"language":"ru","status":"unofficial","text":"013K (1): в абелевой категории с достаточным запасом инъективных у всякого объекта есть инъективная резольвента"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (1): if \\(RF\\) is everywhere defined, then \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): если \\(RF\\) определён всюду, то \\(R^iF = 0\\) при \\(i < 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (3): if \\(RF\\) is everywhere defined and \\(F\\) is left exact, then \\(F → R^0F\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05TD (3): если \\(RF\\) определён всюду и \\(F\\) точен слева, то \\(F → R^0F\\) — изоморфизм"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RFEverywhereDefined","kind":"rule","labels":[{"language":"en","status":"official","text":"05TI (2): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is an additive functor into an abelian category, then \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05TI (2): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) — аддитивный функтор в абелеву категорию, то \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TE (1): the functors \\(R^iF\\), \\(i ≥ 0\\), carry a canonical structure of a cohomological \\(δ\\)-functor (010Q)"},{"language":"ru","status":"unofficial","text":"05TE (1): функторы \\(R^iF\\), \\(i ≥ 0\\), несут каноническую структуру когомологического \\(δ\\)-функтора (010Q)"}],"name":"derived_functors_form_delta_functor","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#derived_functors_universal","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"015B (4): the sequence \\((R^iF, δ)\\) is a universal \\(δ\\)-functor (010S) from \\(A\\) to \\(B\\)"},{"language":"ru","status":"unofficial","text":"015B (4): последовательность \\((R^iF, δ)\\) — универсальный \\(δ\\)-функтор (010S) из \\(A\\) в \\(B\\)"}],"name":"derived_functors_universal","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_universal/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#has_injective_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the object has an injective resolution"},{"language":"ru","status":"unofficial","text":"у объекта есть инъективная резольвента"}],"name":"has_injective_resolution","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#has_injective_resolution/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#higher_derived_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"015C: \\(R^iF(A) = 0\\) for all \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"015C: \\(R^iF(A) = 0\\) при всех \\(i > 0\\)"}],"name":"higher_derived_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#higher_derived_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"urn:stacks:clir:derived-functors#higher_derived_vanish/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (1): \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): \\(R^iF = 0\\) при \\(i < 0\\)"}],"name":"negative_derived_functors_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#r0_agrees_with_f","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (3): the map \\(F → R^0F\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05TD (3): отображение \\(F → R^0F\\) — изоморфизм"}],"name":"r0_agrees_with_f","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#r0_agrees_with_f/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SV: the right derived functor \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05SV: правый производный функтор \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"name":"rf_everywhere_defined","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#right_acyclic_for","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0157 (3): the object \\(A\\) is right acyclic for \\(F\\): \\(A[0]\\) computes \\(RF\\)"},{"language":"ru","status":"unofficial","text":"0157 (3): объект \\(A\\) правый ацикличный для \\(F\\): \\(A[0]\\) вычисляет \\(RF\\)"}],"name":"right_acyclic_for","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#right_acyclic_for/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:derived-functors#Obj"}},{"id":"urn:stacks:clir:derived-functors#right_acyclic_for/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_013K","urn:stacks:clir:derived-functors#ST_015B","urn:stacks:clir:derived-functors#ST_015C","urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4","urn:stacks:clir:derived-functors#ST_05TD","urn:stacks:clir:derived-functors#ST_05TE","urn:stacks:clir:derived-functors#ST_05TH","urn:stacks:clir:derived-functors#ST_05TI"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные 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государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317","resultHash":"sha256:948ded4c9addb13dfd2ef08b0a73e97303a07d94cae1b83b553c0a82e679e515","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:f","urn:case:stacks:j"],"facts":[{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"enough_injectives"},{"args":["urn:case:stacks:f","urn:case:stacks:cat-a","urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"functor_between"},{"args":["urn:case:stacks:f"],"package":"stacks-categories","predicate":"preserves_left_exactness"},{"args":["urn:case:stacks:j","urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"object_of"},{"args":["urn:case:stacks:j"],"package":"stacks-categories","predicate":"lifting_property"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-derived-functors","predicate":"higher_derived_vanish","proof":true},"sources":[{"contentHash":"sha256:9698ad04b03a0ca888a87e8f88e9925a856764046ae30cfb569730f33de5c92d","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_013K","kind":"fragment","locator":"tag/013K","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:869cffaf0235a6f9d2d38c26d042d9ff1c0bbfae733cd1cdcc72af28b5bc8fa4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-injective-resolutions-exist}\nLet $\\mathcal{A}$ be an abelian category.\nAssume $\\mathcal{A}$ has enough injectives.\n\\begin{enumerate}\n\\item Any object of $\\mathcal{A}$ has an injective resolution.\n\\item If $H^n(K^\\bullet) = 0$ for all $n \\ll 0$ then\n$K^\\bullet$ has an injective resolution.\n\\item If $K^\\bullet$ is a complex with $K^n = 0$ for $n < a$, then\nthere exists an injective resolution $\\alpha : K^\\bullet \\to I^\\bullet$\nwith $I^n = 0$ for $n < a$ such that each $\\alpha^n : K^n \\to I^n$ is\ninjective.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:4afa5f4b9810c92dab35571346daba32fe103025b5c3cd8088dbd7e54ecd20fe","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_015B","kind":"fragment","locator":"tag/015B","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:dfebd11201136d7893be34773ff7fda188a8b1936ac0ddeff5a25c3f02c5f5a0","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-higher-derived-functors}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.\n\\begin{enumerate}\n\\item For any short exact sequence\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nof complexes in $\\text{Comp}^{+}(\\mathcal{A})$ there\nis an associated long exact sequence\n$$\n\\ldots \\to\nH^i(RF(A^\\bullet)) \\to\nH^i(RF(B^\\bullet)) \\to\nH^i(RF(C^\\bullet)) \\to\nH^{i + 1}(RF(A^\\bullet)) \\to \\ldots\n$$\n\\item The functors $R^iF : \\mathcal{A} \\to \\mathcal{B}$\nare zero for $i < 0$. Also $R^0F = F : \\mathcal{A} \\to \\mathcal{B}$.\n\\item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective.\n\\item The sequence $(R^iF, \\delta)$ forms a universal $\\delta$-functor (see\nHomology, Definition \\ref{homology-definition-universal-delta-functor})\nfrom $\\mathcal{A}$ to $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:e250263a8b2f3813392cdbde3901795ceda78f89b9865dfdc2a7794d781e519a","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_015C","kind":"fragment","locator":"tag/015C","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:d0947fabef80aaa804f6f4635edca13dba8e17d1cdf28aacd25709217ffafa46","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-F-acyclic}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $A$ be an object of $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $A$ is right acyclic for $F$ if and only if\n$F(A) \\to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$,\n\\item if $F$ is left exact, then $A$ is right acyclic for $F$\nif and only if $R^iF(A) = 0$ for all $i > 0$.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SV","kind":"fragment","locator":"tag/05SV","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-everywhere-defined}\nIn\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.\n\\end{definition}"}]},{"contentHash":"sha256:7b734956d4fc9c61b7d92b47efd6278d84ca8bed87f9ecb3d0b1bb74c80f8ea8","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"situation","id":"urn:stacks:clir:derived-functors#ST_05T4","kind":"fragment","locator":"tag/05T4","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6","language":"en","status":"official","text":"\\begin{situation}\n\\label{situation-classical}\nHere $F : \\mathcal{A} \\to \\mathcal{B}$ is an additive functor between\nabelian categories. This induces exact functors\n$$\nF : K(\\mathcal{A}) \\to K(\\mathcal{B}), \\quad\nK^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B}), \\quad\nK^{-}(\\mathcal{A}) \\to K^{-}(\\mathcal{B}).\n$$\nSee Lemma \\ref{lemma-additive-exact-homotopy-category}.\nWe also denote $F$ the composition $K(\\mathcal{A}) \\to D(\\mathcal{B})$,\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$, and\n$K^{-}(\\mathcal{A}) \\to D^-(\\mathcal{B})$ of $F$ with the localization\nfunctor $K(\\mathcal{B}) \\to D(\\mathcal{B})$, etc. This situation leads\nto four derived functors we will consider in the following.\n\\begin{enumerate}\n\\item The right derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The right derived functor of\n$F : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^{+}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^-(\\mathcal{A})$.\n\\end{enumerate}\nEach of these cases is an example of\nSituation \\ref{situation-derived-functor}.\n\\end{situation}"}]},{"contentHash":"sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TD","kind":"fragment","locator":"tag/05TD","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-left-exact-higher-derived}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TE","kind":"fragment","locator":"tag/05TE","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-right-derived-delta-functor}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:67d5bab826525b8b40aa6d952b68667e49a96a051a298a45f2890a4bfb11950e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TH","kind":"fragment","locator":"tag/05TH","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e032a20e4fb0e18232eca1300982b8c867068569810bf8e4474e48073cab1465","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-injective-acyclic}\nLet $\\mathcal{A}$ be an abelian category.\nLet $I \\in \\Ob(\\mathcal{A})$ be an injective object.\nLet $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$\nfor any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto any triangulated category $\\mathcal{D}$.\n\\item $I$ is right acyclic for any additive functor\n$F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TI","kind":"fragment","locator":"tag/05TI","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-enough-injectives-right-derived}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:64c741ef9deb2269f16508f201759a9d9bd1691e833d5213c0a244b78a7ce59a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:categories#ST_010N","kind":"fragment","locator":"tag/010N","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:e66f034eeeacfcb7b11738065188279ef03410097d4c35d3837c698b10ba0cab","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-exact-functor}\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\n\\begin{enumerate}\n\\item If $F$ is either left or right exact, then it is additive.\n\\item $F$ is left exact if and only if\nfor every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C)$\nis exact.\n\\item $F$ is right exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\item $F$ is exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\end{enumerate}\n\\end{lemma}"}],"visibility":"public"},{"contentHash":"sha256:2ae5842758866ecdbe553b178de00c7c3e8ea454e1a32ff6ed155b6671ddb454","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0135","kind":"fragment","locator":"tag/0135","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:3b91442b6a83fe54c0144799446ea4e30d2c23f4f254154a4942fbe3fda8bb46","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-injective}\nLet $\\mathcal{A}$ be an abelian category.\nAn object $J \\in \\Ob(\\mathcal{A})$ is\ncalled {\\it injective} if for every injection\n$A \\hookrightarrow B$ and every morphism\n$A \\to J$ there exists a morphism $B \\to J$ making\nthe following diagram commute\n$$\n\\xymatrix{\nA \\ar[r] \\ar[d] & B \\ar@{-->}[ld] \\\\\nJ &\n}\n$$\n\\end{definition}"}],"visibility":"public"}],"text":"higher_derived_vanish: **TRUE_ONLY** — установлено\nВыведено правом: left_exact_functor(urn:case:stacks:f); additive_functor(urn:case:stacks:f); injective_object(urn:case:stacks:j); has_injective_resolution(urn:case:stacks:j); derived_functors_universal(urn:case:stacks:f); right_acyclic_for(urn:case:stacks:j, urn:case:stacks:f); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); higher_derived_vanish(urn:case:stacks:f, urn:case:stacks:j); negative_derived_functors_vanish(urn:case:stacks:f); r0_agrees_with_f(urn:case:stacks:f)\n…и ещё 4 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: AdditiveByFiniteProducts, InjectiveByLifting, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, HigherDerivedVanishForAcyclic, InjectiveIsRightAcyclic, InjectiveResolutionsExist, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 33 узлов — поле evaluation готово для 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B\\)"},{"language":"ru","status":"unofficial","text":"функтор действует из первой категории во вторую: \\(F : A → B\\)"}],"name":"functor_between","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#functor_between/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Functor"}},{"id":"urn:stacks:clir:category-theory#functor_between/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:category-theory#Category"}},{"id":"urn:stacks:clir:category-theory#functor_between/arg/b","labels":[],"name":"b","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#has_finite_products","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"001S: products \\(x × y\\) exist for all pairs of objects, hence all finite products"},{"language":"ru","status":"unofficial","text":"001S: произведения \\(x × y\\) существуют для всех пар объектов, а значит все конечные произведения"}],"name":"has_finite_products","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#has_finite_products/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#left_exact_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0034 (1): the functor is left exact"},{"language":"ru","status":"unofficial","text":"0034 (1): функтор точен слева"}],"name":"left_exact_functor","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#left_exact_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"05TE (1): if \\(RF\\) is everywhere defined, the \\(R^iF\\) come equipped with a canonical structure of a \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"05TE (1): если \\(RF\\) определён всюду, \\(R^iF\\) снабжены канонической структурой \\(δ\\)-функтора"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"015B (4): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is left exact, then \\((R^iF, δ)\\) is a universal \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"015B (4): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) точен слева, то \\((R^iF, δ)\\) — универсальный \\(δ\\)-функтор"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (1): if \\(RF\\) is everywhere defined, then \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): если \\(RF\\) определён всюду, то \\(R^iF = 0\\) при \\(i < 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (3): if \\(RF\\) is everywhere defined and \\(F\\) is left exact, then \\(F → R^0F\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05TD (3): если \\(RF\\) определён всюду и \\(F\\) точен слева, то \\(F → R^0F\\) — изоморфизм"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RFEverywhereDefined","kind":"rule","labels":[{"language":"en","status":"official","text":"05TI (2): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is an additive functor into an abelian category, then \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05TI (2): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) — аддитивный функтор в абелеву категорию, то \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TE (1): the functors \\(R^iF\\), \\(i ≥ 0\\), carry a canonical structure of a cohomological \\(δ\\)-functor (010Q)"},{"language":"ru","status":"unofficial","text":"05TE (1): функторы \\(R^iF\\), \\(i ≥ 0\\), несут каноническую структуру когомологического \\(δ\\)-функтора (010Q)"}],"name":"derived_functors_form_delta_functor","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#derived_functors_universal","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"015B (4): the sequence \\((R^iF, δ)\\) is a universal \\(δ\\)-functor (010S) from \\(A\\) to \\(B\\)"},{"language":"ru","status":"unofficial","text":"015B (4): последовательность \\((R^iF, δ)\\) — универсальный \\(δ\\)-функтор (010S) из \\(A\\) в \\(B\\)"}],"name":"derived_functors_universal","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_universal/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (1): \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): \\(R^iF = 0\\) при \\(i < 0\\)"}],"name":"negative_derived_functors_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#r0_agrees_with_f","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (3): the map \\(F → R^0F\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05TD (3): отображение \\(F → R^0F\\) — изоморфизм"}],"name":"r0_agrees_with_f","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#r0_agrees_with_f/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SV: the right derived functor \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05SV: правый производный функтор \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"name":"rf_everywhere_defined","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_015B","urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4","urn:stacks:clir:derived-functors#ST_05TD","urn:stacks:clir:derived-functors#ST_05TE","urn:stacks:clir:derived-functors#ST_05TI"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные функторы по The Stacks Project: определения и леммы как словарь с 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доктрина"}],"caseHash":"sha256:6d3ce68db48741c17f22dd5853ca7a372d510f6347d629f3962ee4a413fb458d","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317","resultHash":"sha256:cc8e232a999916a79d443dadb84a6a337eb7749cc48116ea22d80d817700f20e","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:f"],"facts":[{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"enough_injectives"},{"args":["urn:case:stacks:f","urn:case:stacks:cat-a","urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"functor_between"},{"args":["urn:case:stacks:f"],"package":"stacks-categories","predicate":"preserves_left_exactness"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-derived-functors","predicate":"r0_agrees_with_f","proof":true},"sources":[{"contentHash":"sha256:4afa5f4b9810c92dab35571346daba32fe103025b5c3cd8088dbd7e54ecd20fe","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_015B","kind":"fragment","locator":"tag/015B","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:dfebd11201136d7893be34773ff7fda188a8b1936ac0ddeff5a25c3f02c5f5a0","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-higher-derived-functors}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.\n\\begin{enumerate}\n\\item For any short exact sequence\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nof complexes in $\\text{Comp}^{+}(\\mathcal{A})$ there\nis an associated long exact sequence\n$$\n\\ldots \\to\nH^i(RF(A^\\bullet)) \\to\nH^i(RF(B^\\bullet)) \\to\nH^i(RF(C^\\bullet)) \\to\nH^{i + 1}(RF(A^\\bullet)) \\to \\ldots\n$$\n\\item The functors $R^iF : \\mathcal{A} \\to \\mathcal{B}$\nare zero for $i < 0$. Also $R^0F = F : \\mathcal{A} \\to \\mathcal{B}$.\n\\item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective.\n\\item The sequence $(R^iF, \\delta)$ forms a universal $\\delta$-functor (see\nHomology, Definition \\ref{homology-definition-universal-delta-functor})\nfrom $\\mathcal{A}$ to $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SV","kind":"fragment","locator":"tag/05SV","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-everywhere-defined}\nIn\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.\n\\end{definition}"}]},{"contentHash":"sha256:7b734956d4fc9c61b7d92b47efd6278d84ca8bed87f9ecb3d0b1bb74c80f8ea8","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"situation","id":"urn:stacks:clir:derived-functors#ST_05T4","kind":"fragment","locator":"tag/05T4","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6","language":"en","status":"official","text":"\\begin{situation}\n\\label{situation-classical}\nHere $F : \\mathcal{A} \\to \\mathcal{B}$ is an additive functor between\nabelian categories. This induces exact functors\n$$\nF : K(\\mathcal{A}) \\to K(\\mathcal{B}), \\quad\nK^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B}), \\quad\nK^{-}(\\mathcal{A}) \\to K^{-}(\\mathcal{B}).\n$$\nSee Lemma \\ref{lemma-additive-exact-homotopy-category}.\nWe also denote $F$ the composition $K(\\mathcal{A}) \\to D(\\mathcal{B})$,\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$, and\n$K^{-}(\\mathcal{A}) \\to D^-(\\mathcal{B})$ of $F$ with the localization\nfunctor $K(\\mathcal{B}) \\to D(\\mathcal{B})$, etc. This situation leads\nto four derived functors we will consider in the following.\n\\begin{enumerate}\n\\item The right derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The right derived functor of\n$F : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^{+}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^-(\\mathcal{A})$.\n\\end{enumerate}\nEach of these cases is an example of\nSituation \\ref{situation-derived-functor}.\n\\end{situation}"}]},{"contentHash":"sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TD","kind":"fragment","locator":"tag/05TD","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-left-exact-higher-derived}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TE","kind":"fragment","locator":"tag/05TE","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-right-derived-delta-functor}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TI","kind":"fragment","locator":"tag/05TI","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-enough-injectives-right-derived}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:64c741ef9deb2269f16508f201759a9d9bd1691e833d5213c0a244b78a7ce59a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:categories#ST_010N","kind":"fragment","locator":"tag/010N","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:e66f034eeeacfcb7b11738065188279ef03410097d4c35d3837c698b10ba0cab","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-exact-functor}\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\n\\begin{enumerate}\n\\item If $F$ is either left or right exact, then it is additive.\n\\item $F$ is left exact if and only if\nfor every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C)$\nis exact.\n\\item $F$ is right exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\item $F$ is exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\end{enumerate}\n\\end{lemma}"}],"visibility":"public"}],"text":"r0_agrees_with_f: **TRUE_ONLY** — установлено\nВыведено правом: left_exact_functor(urn:case:stacks:f); additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); derived_functors_universal(urn:case:stacks:f); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f); r0_agrees_with_f(urn:case:stacks:f)\nПрименены правила: AdditiveByFiniteProducts, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 27 узлов — поле evaluation готово для 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→ B → C → 0\\) the sequence \\(0 → F(A) → F(B) → F(C)\\) is exact"},{"language":"ru","status":"unofficial","text":"для всякой короткой точной последовательности \\(0 → A → B → C → 0\\) последовательность \\(0 → F(A) → F(B) → F(C)\\) точна"}],"name":"preserves_left_exactness","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#preserves_left_exactness/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:categories#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#short_exact_sequence","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the sequence \\(0 → A → B → C → 0\\) is a short exact sequence in the category"},{"language":"ru","status":"unofficial","text":"последовательность \\(0 → A → B → C → 0\\) — короткая точная последовательность в категории"}],"name":"short_exact_sequence","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#short_exact_sequence/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:categories#ShortExactSequence"}},{"id":"urn:stacks:clir:categories#short_exact_sequence/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#Category","kind":"type_decl","labels":[{"language":"en","status":"official","text":"category"},{"language":"ru","status":"unofficial","text":"категория"}],"name":"Category","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#Functor","kind":"type_decl","labels":[{"language":"en","status":"official","text":"functor"},{"language":"ru","status":"unofficial","text":"функтор"}],"name":"Functor","package":"urn:stacks:clir:category-theory"},{"id":"urn:stacks:clir:category-theory#functor_between","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor goes from the first category to the second: \\(F : A → B\\)"},{"language":"ru","status":"unofficial","text":"функтор действует из первой категории во вторую: \\(F : A → B\\)"}],"name":"functor_between","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#functor_between/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Functor"}},{"id":"urn:stacks:clir:category-theory#functor_between/arg/a","labels":[],"name":"a","type":{"name":"urn:stacks:clir:category-theory#Category"}},{"id":"urn:stacks:clir:category-theory#functor_between/arg/b","labels":[],"name":"b","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#has_finite_products","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"001S: products \\(x × y\\) exist for all pairs of objects, hence all finite products"},{"language":"ru","status":"unofficial","text":"001S: произведения \\(x × y\\) существуют для всех пар объектов, а значит все конечные произведения"}],"name":"has_finite_products","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#has_finite_products/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:category-theory#left_exact_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0034 (1): the functor is left exact"},{"language":"ru","status":"unofficial","text":"0034 (1): функтор точен слева"}],"name":"left_exact_functor","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#left_exact_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"05TE (1): if \\(RF\\) is everywhere defined, the \\(R^iF\\) come equipped with a canonical structure of a \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"05TE (1): если \\(RF\\) определён всюду, \\(R^iF\\) снабжены канонической структурой \\(δ\\)-функтора"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"015B (4): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is left exact, then \\((R^iF, δ)\\) is a universal \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"015B (4): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) точен слева, то \\((R^iF, δ)\\) — универсальный \\(δ\\)-функтор"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#LongExactSequence","kind":"rule","labels":[{"language":"en","status":"official","text":"015B (1): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is left exact, every short exact sequence in \\(A\\) has an associated long exact sequence of the \\(R^iF\\)"},{"language":"ru","status":"unofficial","text":"015B (1): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) точен слева, всякой короткой точной последовательности в \\(A\\) отвечает длинная точная последовательность \\(R^iF\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (1): if \\(RF\\) is everywhere defined, then \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): если \\(RF\\) определён всюду, то \\(R^iF = 0\\) при \\(i < 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (3): if \\(RF\\) is everywhere defined and \\(F\\) is left exact, then \\(F → R^0F\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05TD (3): если \\(RF\\) определён всюду и \\(F\\) точен слева, то \\(F → R^0F\\) — изоморфизм"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RFEverywhereDefined","kind":"rule","labels":[{"language":"en","status":"official","text":"05TI (2): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is an additive functor into an abelian category, then \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05TI (2): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) — аддитивный функтор в абелеву категорию, то \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TE (1): the functors \\(R^iF\\), \\(i ≥ 0\\), carry a canonical structure of a cohomological \\(δ\\)-functor (010Q)"},{"language":"ru","status":"unofficial","text":"05TE (1): функторы \\(R^iF\\), \\(i ≥ 0\\), несут каноническую структуру когомологического \\(δ\\)-функтора (010Q)"}],"name":"derived_functors_form_delta_functor","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#derived_functors_universal","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"015B (4): the sequence \\((R^iF, δ)\\) is a universal \\(δ\\)-functor (010S) from \\(A\\) to \\(B\\)"},{"language":"ru","status":"unofficial","text":"015B (4): последовательность \\((R^iF, δ)\\) — универсальный \\(δ\\)-функтор (010S) из \\(A\\) в \\(B\\)"}],"name":"derived_functors_universal","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_universal/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#long_exact_sequence_of_derived","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"015B (1): the short exact sequence \\(0 → A → B → C → 0\\) has an associated long exact sequence \\(… → R^iF(A) → R^iF(B) → R^iF(C) → R^{i+1}F(A) → …\\)"},{"language":"ru","status":"unofficial","text":"015B (1): короткой точной последовательности \\(0 → A → B → C → 0\\) отвечает длинная точная последовательность \\(… → R^iF(A) → R^iF(B) → R^iF(C) → R^{i+1}F(A) → …\\)"}],"name":"long_exact_sequence_of_derived","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#long_exact_sequence_of_derived/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"urn:stacks:clir:derived-functors#long_exact_sequence_of_derived/arg/s","labels":[],"name":"s","type":{"name":"urn:stacks:clir:derived-functors#ShortExactSequence"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (1): \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): \\(R^iF = 0\\) при \\(i < 0\\)"}],"name":"negative_derived_functors_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#r0_agrees_with_f","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (3): the map \\(F → R^0F\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"05TD (3): отображение \\(F → R^0F\\) — изоморфизм"}],"name":"r0_agrees_with_f","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#r0_agrees_with_f/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SV: the right derived functor \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05SV: правый производный функтор \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"name":"rf_everywhere_defined","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_015B","urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4","urn:stacks:clir:derived-functors#ST_05TD","urn:stacks:clir:derived-functors#ST_05TE","urn:stacks:clir:derived-functors#ST_05TI"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные функторы по The Stacks Project: определения и леммы как словарь с 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$\\mathcal{A}$ be an abelian category with enough injectives.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a left exact functor.\n\\begin{enumerate}\n\\item For any short exact sequence\n$0 \\to A^\\bullet \\to B^\\bullet \\to C^\\bullet \\to 0$\nof complexes in $\\text{Comp}^{+}(\\mathcal{A})$ there\nis an associated long exact sequence\n$$\n\\ldots \\to\nH^i(RF(A^\\bullet)) \\to\nH^i(RF(B^\\bullet)) \\to\nH^i(RF(C^\\bullet)) \\to\nH^{i + 1}(RF(A^\\bullet)) \\to \\ldots\n$$\n\\item The functors $R^iF : \\mathcal{A} \\to \\mathcal{B}$\nare zero for $i < 0$. Also $R^0F = F : \\mathcal{A} \\to \\mathcal{B}$.\n\\item We have $R^iF(I) = 0$ for $i > 0$ and $I$ injective.\n\\item The sequence $(R^iF, \\delta)$ forms a universal $\\delta$-functor (see\nHomology, Definition \\ref{homology-definition-universal-delta-functor})\nfrom $\\mathcal{A}$ to $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SV","kind":"fragment","locator":"tag/05SV","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-everywhere-defined}\nIn\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.\n\\end{definition}"}]},{"contentHash":"sha256:7b734956d4fc9c61b7d92b47efd6278d84ca8bed87f9ecb3d0b1bb74c80f8ea8","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"situation","id":"urn:stacks:clir:derived-functors#ST_05T4","kind":"fragment","locator":"tag/05T4","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6","language":"en","status":"official","text":"\\begin{situation}\n\\label{situation-classical}\nHere $F : \\mathcal{A} \\to \\mathcal{B}$ is an additive functor between\nabelian categories. This induces exact functors\n$$\nF : K(\\mathcal{A}) \\to K(\\mathcal{B}), \\quad\nK^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B}), \\quad\nK^{-}(\\mathcal{A}) \\to K^{-}(\\mathcal{B}).\n$$\nSee Lemma \\ref{lemma-additive-exact-homotopy-category}.\nWe also denote $F$ the composition $K(\\mathcal{A}) \\to D(\\mathcal{B})$,\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$, and\n$K^{-}(\\mathcal{A}) \\to D^-(\\mathcal{B})$ of $F$ with the localization\nfunctor $K(\\mathcal{B}) \\to D(\\mathcal{B})$, etc. This situation leads\nto four derived functors we will consider in the following.\n\\begin{enumerate}\n\\item The right derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The right derived functor of\n$F : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^{+}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^-(\\mathcal{A})$.\n\\end{enumerate}\nEach of these cases is an example of\nSituation \\ref{situation-derived-functor}.\n\\end{situation}"}]},{"contentHash":"sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TD","kind":"fragment","locator":"tag/05TD","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-left-exact-higher-derived}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TE","kind":"fragment","locator":"tag/05TE","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-right-derived-delta-functor}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TI","kind":"fragment","locator":"tag/05TI","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-enough-injectives-right-derived}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:64c741ef9deb2269f16508f201759a9d9bd1691e833d5213c0a244b78a7ce59a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:categories#ST_010N","kind":"fragment","locator":"tag/010N","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:e66f034eeeacfcb7b11738065188279ef03410097d4c35d3837c698b10ba0cab","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-exact-functor}\nLet $\\mathcal{A}$ and $\\mathcal{B}$ be abelian categories.\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be a functor.\n\\begin{enumerate}\n\\item If $F$ is either left or right exact, then it is additive.\n\\item $F$ is left exact if and only if\nfor every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C)$\nis exact.\n\\item $F$ is right exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\item $F$ is exact if and only if for every short exact sequence\n$0 \\to A \\to B \\to C \\to 0$\nthe sequence $0 \\to F(A) \\to F(B) \\to F(C) \\to 0$\nis exact.\n\\end{enumerate}\n\\end{lemma}"}],"visibility":"public"}],"text":"long_exact_sequence_of_derived: **TRUE_ONLY** — установлено\nВыведено правом: left_exact_functor(urn:case:stacks:f); additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); derived_functors_universal(urn:case:stacks:f); long_exact_sequence_of_derived(urn:case:stacks:f, urn:case:stacks:s); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f); r0_agrees_with_f(urn:case:stacks:f)\nПрименены правила: AdditiveByFiniteProducts, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, LongExactSequence, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 29 узлов — поле evaluation готово для 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\\(J\\)"}],"name":"enough_injectives","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#enough_injectives/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_cokernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all cokernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все коядра"}],"name":"has_all_cokernels","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#has_all_cokernels/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#has_all_kernels","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"all kernels exist in the category"},{"language":"ru","status":"unofficial","text":"в категории существуют все 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y\\)"}],"name":"homomorphism_on_hom_groups","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#homomorphism_on_hom_groups/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:categories#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#preadditive_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"00ZY: every \\(Mor(x, y)\\) is an abelian group and composition is bilinear"},{"language":"ru","status":"unofficial","text":"00ZY: каждое \\(Mor(x, y)\\) — абелева группа, композиция 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а значит все конечные произведения"}],"name":"has_finite_products","package":"urn:stacks:clir:category-theory","parameters":[{"id":"urn:stacks:clir:category-theory#has_finite_products/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:category-theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"05TE (1): if \\(RF\\) is everywhere defined, the \\(R^iF\\) come equipped with a canonical structure of a \\(δ\\)-functor"},{"language":"ru","status":"unofficial","text":"05TE (1): если \\(RF\\) определён всюду, \\(R^iF\\) снабжены канонической структурой \\(δ\\)-функтора"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","kind":"rule","labels":[{"language":"en","status":"official","text":"05TD (1): if \\(RF\\) is everywhere defined, then \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): если \\(RF\\) определён всюду, то \\(R^iF = 0\\) при \\(i < 0\\)"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#RFEverywhereDefined","kind":"rule","labels":[{"language":"en","status":"official","text":"05TI (2): if \\(A\\) is abelian with enough injectives and \\(F : A → B\\) is an additive functor into an abelian category, then \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05TI (2): если \\(A\\) абелева с достаточным запасом инъективных и \\(F : A → B\\) — аддитивный функтор в абелеву категорию, то \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"package":"urn:stacks:clir:derived-functors","strength":"strict"},{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TE (1): the functors \\(R^iF\\), \\(i ≥ 0\\), carry a canonical structure of a cohomological \\(δ\\)-functor (010Q)"},{"language":"ru","status":"unofficial","text":"05TE (1): функторы \\(R^iF\\), \\(i ≥ 0\\), несут каноническую структуру когомологического \\(δ\\)-функтора (010Q)"}],"name":"derived_functors_form_delta_functor","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05TD (1): \\(R^iF = 0\\) for \\(i < 0\\)"},{"language":"ru","status":"unofficial","text":"05TD (1): \\(R^iF = 0\\) при \\(i < 0\\)"}],"name":"negative_derived_functors_vanish","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"05SV: the right derived functor \\(RF : D⁺(A) → D⁺(B)\\) is everywhere defined"},{"language":"ru","status":"unofficial","text":"05SV: правый производный функтор \\(RF : D⁺(A) → D⁺(B)\\) определён всюду"}],"name":"rf_everywhere_defined","package":"urn:stacks:clir:derived-functors","parameters":[{"id":"urn:stacks:clir:derived-functors#rf_everywhere_defined/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":16,"fragments":["urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4","urn:stacks:clir:derived-functors#ST_05TD","urn:stacks:clir:derived-functors#ST_05TE","urn:stacks:clir:derived-functors#ST_05TI"],"jurisdiction":"none","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","title":"Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_00ZY","urn:stacks:clir:categories#ST_0104","urn:stacks:clir:categories#ST_0109"],"jurisdiction":"none","namespace":"urn:stacks:clir:categories","package":"stacks-categories","title":"Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина"},{"contributed":false,"fragmentCount":17,"fragments":[],"jurisdiction":"none","namespace":"urn:stacks:clir:category-theory","package":"stacks-category-theory","title":"Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:7299d7d4001b7f6154f0372b741e9c7898ae828311ac9148a175f4d580767b11","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317","resultHash":"sha256:64baeb47c951a339c0a4735727d5d4cd6d007f068a19d114c8cfa604bbb2cc17","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:f"],"facts":[{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"preadditive_category"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"has_finite_products"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_kernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"has_all_cokernels"},{"args":["urn:case:stacks:cat-b"],"package":"stacks-categories","predicate":"coimage_to_image_isomorphism"},{"args":["urn:case:stacks:cat-a"],"package":"stacks-categories","predicate":"enough_injectives"},{"args":["urn:case:stacks:f","urn:case:stacks:cat-a","urn:case:stacks:cat-b"],"package":"stacks-category-theory","predicate":"functor_between"},{"args":["urn:case:stacks:f"],"package":"stacks-categories","predicate":"homomorphism_on_hom_groups"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-derived-functors","predicate":"negative_derived_functors_vanish","proof":true},"sources":[{"contentHash":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:derived-functors#ST_05SV","kind":"fragment","locator":"tag/05SV","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:c124773f22d16f9d536cdae2abb40060d15bf3c1c27bb039bd7c07afbfc95e45","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-everywhere-defined}\nIn\nSituation \\ref{situation-derived-functor}.\nWe say $F$ is {\\it right derivable}, or that $RF$ {\\it everywhere defined}\nif $RF$ is defined at every object of $\\mathcal{D}$.\nWe say $F$ is {\\it left derivable}, or that $LF$ {\\it everywhere defined}\nif $LF$ is defined at every object of $\\mathcal{D}$.\n\\end{definition}"}]},{"contentHash":"sha256:7b734956d4fc9c61b7d92b47efd6278d84ca8bed87f9ecb3d0b1bb74c80f8ea8","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"situation","id":"urn:stacks:clir:derived-functors#ST_05T4","kind":"fragment","locator":"tag/05T4","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6","language":"en","status":"official","text":"\\begin{situation}\n\\label{situation-classical}\nHere $F : \\mathcal{A} \\to \\mathcal{B}$ is an additive functor between\nabelian categories. This induces exact functors\n$$\nF : K(\\mathcal{A}) \\to K(\\mathcal{B}), \\quad\nK^{+}(\\mathcal{A}) \\to K^{+}(\\mathcal{B}), \\quad\nK^{-}(\\mathcal{A}) \\to K^{-}(\\mathcal{B}).\n$$\nSee Lemma \\ref{lemma-additive-exact-homotopy-category}.\nWe also denote $F$ the composition $K(\\mathcal{A}) \\to D(\\mathcal{B})$,\n$K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$, and\n$K^{-}(\\mathcal{A}) \\to D^-(\\mathcal{B})$ of $F$ with the localization\nfunctor $K(\\mathcal{B}) \\to D(\\mathcal{B})$, etc. This situation leads\nto four derived functors we will consider in the following.\n\\begin{enumerate}\n\\item The right derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The right derived functor of\n$F : K^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^{+}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K(\\mathcal{A}) \\to D(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}(\\mathcal{A})$.\n\\item The left derived functor of\n$F : K^{-}(\\mathcal{A}) \\to D^{-}(\\mathcal{B})$\nrelative to the multiplicative system $\\text{Qis}^-(\\mathcal{A})$.\n\\end{enumerate}\nEach of these cases is an example of\nSituation \\ref{situation-derived-functor}.\n\\end{situation}"}]},{"contentHash":"sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TD","kind":"fragment","locator":"tag/05TD","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:e13afb6775b69471bfd612155b1855cc310461519f6e5d3727e4e64b18e7e5b7","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-left-exact-higher-derived}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined.\n\\begin{enumerate}\n\\item We have $R^iF = 0$ for $i < 0$,\n\\item $R^0F$ is left exact,\n\\item the map $F \\to R^0F$ is an isomorphism if and\nonly if $F$ is left exact.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TE","kind":"fragment","locator":"tag/05TE","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:949b05eff66291065b983c92020aa94af9ab868feaf33ac9dcca7776457d66f4","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-right-derived-delta-functor}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere defined.\n\\begin{enumerate}\n\\item The functors $R^iF$, $i \\geq 0$ come equipped with a canonical\nstructure of a $\\delta$-functor from $\\mathcal{A} \\to \\mathcal{B}$, see\nHomology, Definition \\ref{homology-definition-cohomological-delta-functor}.\n\\item If every object of $\\mathcal{A}$ is a subobject of a right\nacyclic object for $F$, then $\\{R^iF, \\delta\\}_{i \\geq 0}$ is a\nuniversal $\\delta$-functor, see\nHomology, Definition \\ref{homology-definition-universal-delta-functor}.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","edition":"urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:derived-functors#ST_05TI","kind":"fragment","locator":"tag/05TI","package":"urn:stacks:clir:derived-functors","texts":[{"contentHash":"sha256:4d65e3bfe708a80785f85bd358981a9258f467f4705b132e3801b1a913a96651","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-enough-injectives-right-derived}\nLet $\\mathcal{A}$ be an abelian category with enough injectives.\n\\begin{enumerate}\n\\item For any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto a triangulated category $\\mathcal{D}$ the right derived\nfunctor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow \\mathcal{D}\n$$\nis everywhere defined.\n\\item For any additive functor $F : \\mathcal{A} \\to \\mathcal{B}$ into an\nabelian category $\\mathcal{B}$ the right derived functor\n$$\nRF : D^{+}(\\mathcal{A}) \\longrightarrow D^{+}(\\mathcal{B})\n$$\nis everywhere defined.\n\\end{enumerate}\n\\end{lemma}"}]},{"contentHash":"sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_00ZY","kind":"fragment","locator":"tag/00ZY","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0104","kind":"fragment","locator":"tag/0104","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"}],"visibility":"public"},{"contentHash":"sha256:d71e719b696d97622b2d93bda5c18b2271f646d474bfb4ccf2b8276a37bd097d","edition":"urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:categories#ST_0109","kind":"fragment","locator":"tag/0109","package":"urn:stacks:clir:categories","texts":[{"contentHash":"sha256:71bf82d61770e85387a19fb70b7ddfc9fe2d7fffb4dec5a17199db100c9dae22","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-category}\nA category $\\mathcal{A}$ is {\\it abelian} if\nit is additive, if all kernels and cokernels exist,\nand if the natural map $\\Coim(f) \\to \\Im(f)$\nis an isomorphism for all morphisms $f$ of\n$\\mathcal{A}$.\n\\end{definition}"}],"visibility":"public"}],"text":"negative_derived_functors_vanish: **TRUE_ONLY** — установлено\nВыведено правом: additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f)\nПрименены правила: AdditiveByFiniteProducts, AdditiveByHomGroups, abelian_category/sufficient, DerivedFormDeltaFunctor, NegativeDerivedVanish, RFEverywhereDefined\nПраво (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…)\nВместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина; Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина\nproof-граф: 24 узлов — поле evaluation готово для law_explain"}],"language":"ru","question":{"origin":"user","text":"При каких условиях правый производный функтор определён на всей категории, когда R⁰F совпадает с F и почему инъективный объект ацикличен?"},"schemaVersion":"law.answers.document/0.1"}