{"assistant":{"explanations":[{"calculationRefs":["sheafification"],"text":"Из предпучка строится пучок: конструкция сохраняет структуру абелевой группы."},{"calculationRefs":["universality"],"text":"Пучкование универсально среди отображений предпучка в пучки: через него пропускается любое такое отображение."},{"calculationRefs":["h0"],"text":"Функтор глобальных сечений точен слева, поэтому нулевая степень его производного равна ему самому."},{"calculationRefs":["flasque-vanishing"],"text":"Вялость — достаточное условие ацикличности: в положительных степенях когомологий нет."},{"calculationRefs":["injective-flasque"],"text":"Инъективность влечёт вялость, поэтому инъективные резольвенты состоят из ацикличных пучков."},{"calculationRefs":["godement"],"text":"Резольвента Годемана строится для любого пучка, и вычисление когомологий всегда имеет вход."}],"origin":"assistant","summary":"Пучкование предпучка даёт пучок абелевых групп и универсально среди отображений в пучки. Нулевая степень производного функтора глобальных сечений совпадает с самими глобальными сечениями, потому что этот функтор точен слева. У вялого пучка все когомологии положительной степени обращаются в нуль, инъективный пучок вял, а резольвента Годемана существует у любого пучка — поэтому вычислять когомологии есть чем."},"calculations":[{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","sheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","abelian_sheaf_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","has_flasque_resolution(urn:case:stacks:sh:fsharp)","universal_among_maps_to_sheaves(urn:case:stacks:sh:f, urn:case:stacks:sh:fsharp)"],"derivedOmitted":0,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-2"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"definition":{"concept":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"evidence":[],"id":"urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-3"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#plus_construction"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"evidence":[],"id":"urn:proof:apply:SheafificationIsSheaf:eaf6a9aa2d0b4a500dc473cfcc0200ff61ea9beb9b8b4c6f79c6657450c8afd0","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"evidence":[],"id":"urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a","kind":"rule_application","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"evidence":[],"id":"urn:proof:apply:GodementResolutionExists:c3529a34c74fd31d4c672c809a4ecb778e70d8af01b9f392fdbc88127dbc32ff","kind":"rule_application","premises":["urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a"],"rule":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"evidence":[],"id":"urn:proof:apply:SheafifyUniversal:08e13163bd01f4b2580ae4fbd1215e62a1d2eacd896fef56d26ee23b484b96a7","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4","kind":"constraint_check","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae","urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a"],"sourceAnchors":[]}],"proofHash":"sha256:19ec6d489113976ca6298af1f6aad1cb43475c6c9928bca7f8cec72b17567d90","roots":["urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4","urn:proof:query:mcp"]},"resultHash":"sha256:a7ecf5f0ad9b5b795a1992270606048d2aee076e31f2a295d95616e9a1bac8ba","schemaVersion":"law.core.evaluation/0.2"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_007Y","urn:stacks:clir:sheaf-cohomology#ST_0080","urn:stacks:clir:sheaf-cohomology#ST_0085","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:a28ce816820b92ab076db604e2ee24912a3cca06c797faeccfc0a6cf2249c6c3","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e","resultHash":"sha256:a7ecf5f0ad9b5b795a1992270606048d2aee076e31f2a295d95616e9a1bac8ba","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"rulesApplied":["urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf","urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf","urn:stacks:clir:sheaf-cohomology#SheafifyUniversal","urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient"],"signature":{"constants":{},"parameters":[{"labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Obj"}},{"labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on","schemaVersion":"law.answers.signature/0.1","types":{"urn:stacks:clir:category-theory#Obj":{"kind":"unknown"},"urn:stacks:clir:sheaf-cohomology#Space":{"kind":"entity","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks.sheaf_cohomology"}},"vocab":{}},"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf whose compatible families of sections do not glue is not a sheaf","missing":["compatible_sections_glue(urn:case:stacks:sh:fsharp)","presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"],"premises":[{"premise":"compatible_sections_glue(urn:case:stacks:sh:fsharp)","status":"NEITHER"},{"premise":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByFailedGluing"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf in which a glued section is not unique is not a sheaf","missing":["gluing_is_unique(urn:case:stacks:sh:fsharp)","presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"],"premises":[{"premise":"gluing_is_unique(urn:case:stacks:sh:fsharp)","status":"NEITHER"},{"premise":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByFailedUniqueness"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","missing":["not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"],"premises":[{"premise":"not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByRefutingCovering"}],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"artifactHash\":\"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99\",\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:a28ce816820b92ab076db604e2ee24912a3cca06c797faeccfc0a6cf2249c6c3\",\"decisionTime\":\"2026-09-06T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-09-06T12:00:00+05:00\",\"legalTime\":\"2026-09-06\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:6d4ee2eb96c4090c5c0aa2a79391c88683933f7e0b93dade8e7bf518480e49e3\",\"semantics\":\"law.core/0.2\",\"theoryHash\":\"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-2\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_group_structure\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-2\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-3\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#plus_construction\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-3\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafificationIsSheaf:eaf6a9aa2d0b4a500dc473cfcc0200ff61ea9beb9b8b4c6f79c6657450c8afd0\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:GodementResolutionExists:c3529a34c74fd31d4c672c809a4ecb778e70d8af01b9f392fdbc88127dbc32ff\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafifyUniversal:08e13163bd01f4b2580ae4fbd1215e62a1d2eacd896fef56d26ee23b484b96a7\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae\",\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"literal\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\"},\"truthStatus\":\"TRUE_ONLY\"},\"evidence\":[],\"id\":\"urn:proof:query:mcp\",\"kind\":\"query_evaluation\",\"premises\":[\"urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a\"],\"sourceAnchors\":[]}],\"proofHash\":\"sha256:19ec6d489113976ca6298af1f6aad1cb43475c6c9928bca7f8cec72b17567d90\",\"roots\":[\"urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4\",\"urn:proof:query:mcp\"]},\"resultHash\":\"sha256:a7ecf5f0ad9b5b795a1992270606048d2aee076e31f2a295d95616e9a1bac8ba\",\"results\":[{\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:mcp\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatusSupports\":[],\"proof\":\"urn:proof:query:mcp\",\"query\":\"urn:query:mcp\",\"resultKind\":\"PROPOSITION\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_presheaf_on/necessary:f94ce9c98da027a9\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.2\"}","evaluationSha256":"sha256:660311d3d2095dac745534a21a0f23d25eee3b82a4c3839e12ec3e0ae349d7ad","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-2","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-3","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#plus_construction"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"}],"context":{"decisionTime":"2026-09-06T12:00:00+05:00","knowledgeTime":"2026-09-06T12:00:00+05:00","legalTime":"2026-09-06","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:63cba186192426450399328e45e57b3ee07713867431d6f26c106fd5a6bae87e","kind":"world_ref","nodeCount":181,"packages":[{"artifactHash":"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","semanticHash":"sha256:1ee2443e666f565715a3e3a5663914219d324c205aa6004f03c007d582687303"}],"programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.2"},"requestCanonicalSha256":"sha256:e1401480563c107ab3f789156f9c32d91af5df7067072f3bf5831baded03fb54"},"id":"sheafification","kind":"truth","label":"Пучкование даёт пучок абелевых групп","presentation":{"blockers":{"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf whose compatible families of sections do not glue is not a sheaf","premises":[{"args":["urn:case:stacks:sh:fsharp"],"missing":true,"predicate":"compatible_sections_glue","predicateId":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","status":"NEITHER","text":"compatible_sections_glue(urn:case:stacks:sh:fsharp)"},{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"missing":true,"predicate":"presheaf_of_sets_on","predicateId":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","status":"NEITHER","text":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"}],"rule":"NotSheafByFailedGluing","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf in which a glued section is not unique is not a sheaf","premises":[{"args":["urn:case:stacks:sh:fsharp"],"missing":true,"predicate":"gluing_is_unique","predicateId":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","status":"NEITHER","text":"gluing_is_unique(urn:case:stacks:sh:fsharp)"},{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"missing":true,"predicate":"presheaf_of_sets_on","predicateId":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","status":"NEITHER","text":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"}],"rule":"NotSheafByFailedUniqueness","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedUniqueness"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","premises":[{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"missing":true,"predicate":"not_a_sheaf","predicateId":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","status":"NEITHER","text":"not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"}],"rule":"NotSheafByRefutingCovering","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering"}]},"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:e1401480563c107ab3f789156f9c32d91af5df7067072f3bf5831baded03fb54"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"id":"urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006K"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#plus_construction"},"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v2"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"id":"urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0085"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"stacks.category_theory#Obj"}},{"id":"v2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a"],"trust":"engine"}],"symbols":[{"id":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","kind":"rule","labels":[{"language":"en","status":"official","text":"0FKS with 01AD: every abelian sheaf \\(F\\) on \\(X\\) has the Godement resolution \\(0 → F → f_*f^*F → …\\) by flasque sheaves"},{"language":"ru","status":"unofficial","text":"0FKS с 01AD: у всякого абелева пучка \\(F\\) на \\(X\\) есть резольвента Годемана \\(0 → F → f_*f^*F → …\\) вялыми пучками"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf whose compatible families of sections do not glue is not a sheaf"},{"language":"ru","status":"unofficial","text":"006T: предпучок, у которого согласованные семейства сечений не склеиваются, — не пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedUniqueness","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf in which a glued section is not unique is not a sheaf"},{"language":"ru","status":"unofficial","text":"006T: предпучок, в котором склеенное сечение не единственно, — не пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"006T: предпучок, опровергнутый открытым покрытием, — не пучок множеств на \\(X\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf","kind":"rule","labels":[{"language":"en","status":"official","text":"007Y: the presheaf \\(F^#\\) is a sheaf"},{"language":"ru","status":"unofficial","text":"007Y: предпучок \\(F^#\\) — пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf","kind":"rule","labels":[{"language":"en","status":"official","text":"0085: for an abelian presheaf \\(F\\) there is a unique abelian sheaf structure on \\(F^#\\) making \\(F → F^#\\) a morphism of abelian presheaves"},{"language":"ru","status":"unofficial","text":"0085: для абелева предпучка \\(F\\) на \\(F^#\\) есть единственная структура абелева пучка, при которой \\(F → F^#\\) — морфизм абелевых предпучков"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal","kind":"rule","labels":[{"language":"en","status":"official","text":"0080: for a presheaf of sets \\(F\\), any map \\(F → H\\) into a sheaf factors uniquely through \\(F → F^#\\)"},{"language":"ru","status":"unofficial","text":"0080: для предпучка множеств \\(F\\) всякое отображение \\(F → H\\) в пучок единственным образом проходит через \\(F → F^#\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"each \\(F(U)\\) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups"},{"language":"ru","status":"unofficial","text":"каждое \\(F(U)\\) несёт структуру абелевой группы, и все отображения ограничения — гомоморфизмы абелевых групп"}],"name":"abelian_group_structure","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"name":"abelian_presheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0070: \\(F\\) is an abelian sheaf on \\(X\\) — an abelian presheaf whose underlying presheaf of sets is a sheaf"},{"language":"ru","status":"unofficial","text":"0070: \\(F\\) — абелев пучок на \\(X\\): абелев предпучок, чей предпучок множеств — пучок"}],"name":"abelian_sheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, existence: for every open covering \\(U = ∪ U_i\\) and sections \\(s_i ∈ F(U_i)\\) with \\(s_i|U_i∩U_j = s_j|U_i∩U_j\\) there exists \\(s ∈ F(U)\\) with \\(s|U_i = s_i\\)"},{"language":"ru","status":"unofficial","text":"006T, существование: для всякого открытого покрытия \\(U = ∪ U_i\\) и сечений \\(s_i ∈ F(U_i)\\), согласованных на пересечениях, существует \\(s ∈ F(U)\\) с \\(s|U_i = s_i\\)"}],"name":"compatible_sections_glue","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, uniqueness: a section \\(s ∈ F(U)\\) is determined by its restrictions \\(s|U_i\\) to an open covering"},{"language":"ru","status":"unofficial","text":"006T, единственность: сечение \\(s ∈ F(U)\\) определяется своими ограничениями \\(s|U_i\\) на открытое покрытие"}],"name":"gluing_is_unique","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)"},{"language":"ru","status":"unofficial","text":"0FKS: у пучка есть функториальная резольвента вялыми пучками (резольвента Годемана)"}],"name":"has_flasque_resolution","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is not a sheaf on \\(X\\): the sheaf condition 006T fails on some open covering"},{"language":"ru","status":"unofficial","text":"\\(F\\) — не пучок на \\(X\\): условие 006T провалено на некотором открытом покрытии"}],"name":"not_a_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#plus_construction","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(G = F^#\\) is the sheafification of the presheaf \\(F\\): sections over \\(U\\) are compatible families of germs (Sheaves, Section 007X), with the canonical map \\(F → F^#\\)"},{"language":"ru","status":"unofficial","text":"\\(G = F^#\\) — пучковизация предпучка \\(F\\): сечения над \\(U\\) суть согласованные семейства ростков (Sheaves, раздел 007X), с каноническим отображением \\(F → F^#\\)"}],"name":"plus_construction","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#plus_construction/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#plus_construction/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a presheaf of sets on \\(X\\): a rule assigning a set \\(F(U)\\) to each open \\(U\\) and restriction maps \\(ρ^U_V\\) to inclusions \\(V ⊂ U\\) with \\(ρ^U_U = id\\) and \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — предпучок множеств на \\(X\\): правило, сопоставляющее каждому открытому \\(U\\) множество \\(F(U)\\), а вложениям \\(V ⊂ U\\) — отображения ограничения \\(ρ^U_V\\) с \\(ρ^U_U = id\\) и \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"}],"name":"presheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — пучок множеств на \\(X\\)"}],"name":"sheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0080: any map \\(F → H\\) into a sheaf of sets factors uniquely as \\(F → F^# → H\\)"},{"language":"ru","status":"unofficial","text":"0080: всякое отображение \\(F → H\\) в пучок множеств единственным образом раскладывается как \\(F → F^# → H\\)"}],"name":"universal_among_maps_to_sheaves","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_007Y","urn:stacks:clir:sheaf-cohomology#ST_0080","urn:stacks:clir:sheaf-cohomology#ST_0085","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:a28ce816820b92ab076db604e2ee24912a3cca06c797faeccfc0a6cf2249c6c3","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e","resultHash":"sha256:a7ecf5f0ad9b5b795a1992270606048d2aee076e31f2a295d95616e9a1bac8ba","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"facts":[{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"predicate":"presheaf_of_sets_on"},{"args":["urn:case:stacks:sh:f"],"predicate":"abelian_group_structure"},{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:f"],"predicate":"plus_construction"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"abelian_sheaf_on","proof":true},"sources":[{"contentHash":"sha256:fa7c3dec36920848465ed2902237f8efade0877767eebcee37676bb1bb8717d9","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006K","kind":"fragment","locator":"tag/006K","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:b8388b54fb29de485ebd7c91f38c607f6005d41b4d427265a75b163118fd70d6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-presheaves}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf of abelian groups on $X$} or an\n{\\it abelian presheaf over $X$}\nis a presheaf of sets $\\mathcal{F}$ such that for each open\n$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with\nthe structure of an abelian group, and such that all restriction\nmaps $\\rho^U_V$ are homomorphisms of abelian groups, see\nLemma \\ref{lemma-abelian-presheaves} above.\n\\item A {\\it morphism of abelian presheaves over $X$}\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves\nof sets which induces\na homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nfor every open $U \\subset X$.\n\\item The category of presheaves of abelian groups on $X$ is denoted\n$\\textit{PAb}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:8504f30501fb40b30a60685a51aad76a416258206ec9a839891f7bf83cd38323","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006T","kind":"fragment","locator":"tag/006T","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:86546cd976c19579e8056b4fcbae7d781851f1b8799d12d0fba16321efaee59c","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_007Y","kind":"fragment","locator":"tag/007Y","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:282a15f0475c3f68a862ce8346d7972c4f62ab14daa09a1922c02bb9f01c558e","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-sheafification-sheaf}\nThe presheaf $\\mathcal{F}^{\\#}$ is a sheaf.\n\\end{lemma}"}]},{"contentHash":"sha256:9b004386054af3ebceab40302636bc759c8a1b7c732a3984765235c76fc000c4","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0080","kind":"fragment","locator":"tag/0080","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:2ba5dec16073f1fdafcf8a352bfc30f215d20caf060e768c51658875dc939220","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-sheafify-universal}\nLet $\\mathcal{F}$ be a presheaf of sets on $X$.\nAny map $\\mathcal{F} \\to \\mathcal{G}$ into a sheaf of sets\nfactors uniquely as\n$\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$.\n\\end{lemma}"}]},{"contentHash":"sha256:63fafe60d647e15c855184db7359f8c2ead35af2af5d43260bed141fe4d3a8d0","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0085","kind":"fragment","locator":"tag/0085","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:6cfa0751723700062760868c89628b4aa1d8d91b0ca0b69f61af805272569a0e","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-sheafify-abelian-presheaf}\nLet $X$ be a topological space.\nLet $\\mathcal{F}$ be an abelian presheaf on $X$.\nThen there exists a unique structure of\nabelian sheaf on $\\mathcal{F}^\\#$ such that\n$\\mathcal{F} \\to \\mathcal{F}^\\#$ is a morphism\nof abelian presheaves. Moreover, the following adjointness\nproperty holds\n$$\n\\Mor_{\\textit{PAb}(X)}(\\mathcal{F}, i(\\mathcal{G}))\n=\n\\Mor_{\\textit{Ab}(X)}(\\mathcal{F}^\\#, \\mathcal{G}).\n$$\n\\end{lemma}"}]},{"contentHash":"sha256:d0262bf7656850a1933eef167a360437aa2bb6436e57461a3d23de09e8cc8ef3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01AD","kind":"fragment","locator":"tag/01AD","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf","language":"en","status":"official","text":"\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we work out basic notions of sheaves of modules.\nThis in particular includes the case of abelian sheaves, since\nthese may be viewed as sheaves of $\\underline{\\mathbf{Z}}$-modules.\nBasic references are \\cite{FAC}, \\cite{EGA} and \\cite{SGA4}.\n\n\\medskip\\noindent\nWe work out what happens for sheaves of modules on ringed topoi\nin another chapter (see\nModules on Sites, Section \\ref{sites-modules-section-introduction}),\nalthough there we will mostly just duplicate the discussion\nfrom this chapter."}]},{"contentHash":"sha256:787503a0ec46db68e4e11e9bf5fa2fabae3304c806bc7e2f5be69ab1af6cc11f","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0FKS","kind":"fragment","locator":"tag/0FKS","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:8b044713268f8366ca6cc0212d88985ff0517eb2417f464d218f454e93c283fe","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-godement-resolution}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution\n$$\n0 \\to\n\\mathcal{F} \\to\nf_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots\n$$\nfunctorial in $\\mathcal{F}$ such that each term\n$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque\n$\\mathcal{O}_X$-module and such that for all $x \\in X$ the\nmap\n$$\n\\mathcal{F}_x[0] \\to \\Big(\n(f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots\n\\Big)\n$$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.\n\\end{lemma}"}]}],"text":"abelian_sheaf_on: **TRUE_ONLY** — установлено\nВыведено правом: abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); sheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x); abelian_sheaf_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x); has_flasque_resolution(urn:case:stacks:sh:fsharp); universal_among_maps_to_sheaves(urn:case:stacks:sh:f, urn:case:stacks:sh:fsharp)\nПрименены правила: GodementResolutionExists, SheafificationIsSheaf, SheafifyAbelianPresheaf, SheafifyUniversal, abelian_presheaf_on/sufficient\nОтвет поражаем правилом «006T: a presheaf whose compatible families of sections do not glue is not a sheaf» — оно отменило бы вывод, будь установлено: compatible_sections_glue(urn:case:stacks:sh:fsharp); presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)\nОтвет поражаем правилом «006T: a presheaf in which a glued section is not unique is not a sheaf» — оно отменило бы вывод, будь установлено: gluing_is_unique(urn:case:stacks:sh:fsharp); presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)\nОтвет поражаем правилом «006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)» — оно отменило бы вывод, будь установлено: not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:6b62eb59e903…)\nproof-граф: 10 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["sheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","universal_among_maps_to_sheaves(urn:case:stacks:sh:f, urn:case:stacks:sh:fsharp)"],"derivedOmitted":0,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-2"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#plus_construction"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"evidence":[],"id":"urn:proof:apply:SheafificationIsSheaf:af06faaad567ecfebddfcbec65edca5aaad2d911c935bbe8196ed80f8a0b22d2","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"evidence":[],"id":"urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397"],"sourceAnchors":[]}],"proofHash":"sha256:b60fec6a75f8d07e9a748c967d53106bef7e309d53d3a3ecdc9a0b58f36b744a","roots":["urn:proof:query:mcp"]},"resultHash":"sha256:b17365cb02039921b139c2e3dc238b7c0b84788c86a4b6744c37ff68dd75536b","schemaVersion":"law.core.evaluation/0.2"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_007Y","urn:stacks:clir:sheaf-cohomology#ST_0080"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:63d31f683e3fe278f5ccca9a4f909921d7f0367e740eba645a7f56512080c064","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e","resultHash":"sha256:b17365cb02039921b139c2e3dc238b7c0b84788c86a4b6744c37ff68dd75536b","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"rulesApplied":["urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf","urn:stacks:clir:sheaf-cohomology#SheafifyUniversal"],"signature":{"constants":{},"parameters":[{"labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Obj"}},{"labels":[],"name":"g","type":{"name":"urn:stacks:clir:category-theory#Obj"}}],"predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves","schemaVersion":"law.answers.signature/0.1","types":{"urn:stacks:clir:category-theory#Obj":{"kind":"unknown"}},"vocab":{}},"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf whose compatible families of sections do not glue is not a sheaf","missing":["compatible_sections_glue(urn:case:stacks:sh:fsharp)","presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"],"premises":[{"premise":"compatible_sections_glue(urn:case:stacks:sh:fsharp)","status":"NEITHER"},{"premise":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByFailedGluing"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf in which a glued section is not unique is not a sheaf","missing":["gluing_is_unique(urn:case:stacks:sh:fsharp)","presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"],"premises":[{"premise":"gluing_is_unique(urn:case:stacks:sh:fsharp)","status":"NEITHER"},{"premise":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByFailedUniqueness"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","missing":["not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"],"premises":[{"premise":"not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByRefutingCovering"}],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"artifactHash\":\"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99\",\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:63d31f683e3fe278f5ccca9a4f909921d7f0367e740eba645a7f56512080c064\",\"decisionTime\":\"2026-09-06T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-09-06T12:00:00+05:00\",\"legalTime\":\"2026-09-06\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:9a615a73a1376336e2bb4b4e475a713ded831a4f96e256c72dd6847d0551984a\",\"semantics\":\"law.core/0.2\",\"theoryHash\":\"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-2\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#plus_construction\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-2\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafificationIsSheaf:af06faaad567ecfebddfcbec65edca5aaad2d911c935bbe8196ed80f8a0b22d2\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"literal\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:fsharp\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves\"},\"truthStatus\":\"TRUE_ONLY\"},\"evidence\":[],\"id\":\"urn:proof:query:mcp\",\"kind\":\"query_evaluation\",\"premises\":[\"urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397\"],\"sourceAnchors\":[]}],\"proofHash\":\"sha256:b60fec6a75f8d07e9a748c967d53106bef7e309d53d3a3ecdc9a0b58f36b744a\",\"roots\":[\"urn:proof:query:mcp\"]},\"resultHash\":\"sha256:b17365cb02039921b139c2e3dc238b7c0b84788c86a4b6744c37ff68dd75536b\",\"results\":[{\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:mcp\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatusSupports\":[],\"proof\":\"urn:proof:query:mcp\",\"query\":\"urn:query:mcp\",\"resultKind\":\"PROPOSITION\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.2\"}","evaluationSha256":"sha256:bb6b9c8bb5830382dc347c3d5d63d7c38df4cf39775a6d713774db2edbfbb2e9","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-2","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#plus_construction"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"}],"context":{"decisionTime":"2026-09-06T12:00:00+05:00","knowledgeTime":"2026-09-06T12:00:00+05:00","legalTime":"2026-09-06","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:63cba186192426450399328e45e57b3ee07713867431d6f26c106fd5a6bae87e","kind":"world_ref","nodeCount":181,"packages":[{"artifactHash":"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","semanticHash":"sha256:1ee2443e666f565715a3e3a5663914219d324c205aa6004f03c007d582687303"}],"programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.2"},"requestCanonicalSha256":"sha256:8474546a48d586830a2f3b70460fde07d1d669f2aea7b27c369bd7e64de3080c"},"id":"universality","kind":"truth","label":"Универсальность пучкования","presentation":{"blockers":{"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf whose compatible families of sections do not glue is not a sheaf","premises":[{"args":["urn:case:stacks:sh:fsharp"],"missing":true,"predicate":"compatible_sections_glue","predicateId":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","status":"NEITHER","text":"compatible_sections_glue(urn:case:stacks:sh:fsharp)"},{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"missing":true,"predicate":"presheaf_of_sets_on","predicateId":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","status":"NEITHER","text":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"}],"rule":"NotSheafByFailedGluing","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf in which a glued section is not unique is not a sheaf","premises":[{"args":["urn:case:stacks:sh:fsharp"],"missing":true,"predicate":"gluing_is_unique","predicateId":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","status":"NEITHER","text":"gluing_is_unique(urn:case:stacks:sh:fsharp)"},{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"missing":true,"predicate":"presheaf_of_sets_on","predicateId":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","status":"NEITHER","text":"presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"}],"rule":"NotSheafByFailedUniqueness","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedUniqueness"},{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","premises":[{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:x"],"missing":true,"predicate":"not_a_sheaf","predicateId":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","status":"NEITHER","text":"not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"}],"rule":"NotSheafByRefutingCovering","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering"}]},"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:8474546a48d586830a2f3b70460fde07d1d669f2aea7b27c369bd7e64de3080c"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#plus_construction"},"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"head":{"args":[{"kind":"var","var":"v1"},{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"id":"urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0080"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"stacks.category_theory#Obj"}},{"id":"v2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:fsharp","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397"],"trust":"engine"}],"symbols":[{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf whose compatible families of sections do not glue is not a sheaf"},{"language":"ru","status":"unofficial","text":"006T: предпучок, у которого согласованные семейства сечений не склеиваются, — не пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByFailedUniqueness","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf in which a glued section is not unique is not a sheaf"},{"language":"ru","status":"unofficial","text":"006T: предпучок, в котором склеенное сечение не единственно, — не пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"006T: предпучок, опровергнутый открытым покрытием, — не пучок множеств на \\(X\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf","kind":"rule","labels":[{"language":"en","status":"official","text":"007Y: the presheaf \\(F^#\\) is a sheaf"},{"language":"ru","status":"unofficial","text":"007Y: предпучок \\(F^#\\) — пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafifyUniversal","kind":"rule","labels":[{"language":"en","status":"official","text":"0080: for a presheaf of sets \\(F\\), any map \\(F → H\\) into a sheaf factors uniquely through \\(F → F^#\\)"},{"language":"ru","status":"unofficial","text":"0080: для предпучка множеств \\(F\\) всякое отображение \\(F → H\\) в пучок единственным образом проходит через \\(F → F^#\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, existence: for every open covering \\(U = ∪ U_i\\) and sections \\(s_i ∈ F(U_i)\\) with \\(s_i|U_i∩U_j = s_j|U_i∩U_j\\) there exists \\(s ∈ F(U)\\) with \\(s|U_i = s_i\\)"},{"language":"ru","status":"unofficial","text":"006T, существование: для всякого открытого покрытия \\(U = ∪ U_i\\) и сечений \\(s_i ∈ F(U_i)\\), согласованных на пересечениях, существует \\(s ∈ F(U)\\) с \\(s|U_i = s_i\\)"}],"name":"compatible_sections_glue","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, uniqueness: a section \\(s ∈ F(U)\\) is determined by its restrictions \\(s|U_i\\) to an open covering"},{"language":"ru","status":"unofficial","text":"006T, единственность: сечение \\(s ∈ F(U)\\) определяется своими ограничениями \\(s|U_i\\) на открытое покрытие"}],"name":"gluing_is_unique","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is not a sheaf on \\(X\\): the sheaf condition 006T fails on some open covering"},{"language":"ru","status":"unofficial","text":"\\(F\\) — не пучок на \\(X\\): условие 006T провалено на некотором открытом покрытии"}],"name":"not_a_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#plus_construction","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(G = F^#\\) is the sheafification of the presheaf \\(F\\): sections over \\(U\\) are compatible families of germs (Sheaves, Section 007X), with the canonical map \\(F → F^#\\)"},{"language":"ru","status":"unofficial","text":"\\(G = F^#\\) — пучковизация предпучка \\(F\\): сечения над \\(U\\) суть согласованные семейства ростков (Sheaves, раздел 007X), с каноническим отображением \\(F → F^#\\)"}],"name":"plus_construction","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#plus_construction/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#plus_construction/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a presheaf of sets on \\(X\\): a rule assigning a set \\(F(U)\\) to each open \\(U\\) and restriction maps \\(ρ^U_V\\) to inclusions \\(V ⊂ U\\) with \\(ρ^U_U = id\\) and \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — предпучок множеств на \\(X\\): правило, сопоставляющее каждому открытому \\(U\\) множество \\(F(U)\\), а вложениям \\(V ⊂ U\\) — отображения ограничения \\(ρ^U_V\\) с \\(ρ^U_U = id\\) и \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"}],"name":"presheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — пучок множеств на \\(X\\)"}],"name":"sheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0080: any map \\(F → H\\) into a sheaf of sets factors uniquely as \\(F → F^# → H\\)"},{"language":"ru","status":"unofficial","text":"0080: всякое отображение \\(F → H\\) в пучок множеств единственным образом раскладывается как \\(F → F^# → H\\)"}],"name":"universal_among_maps_to_sheaves","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#universal_among_maps_to_sheaves/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_007Y","urn:stacks:clir:sheaf-cohomology#ST_0080"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:63d31f683e3fe278f5ccca9a4f909921d7f0367e740eba645a7f56512080c064","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e","resultHash":"sha256:b17365cb02039921b139c2e3dc238b7c0b84788c86a4b6744c37ff68dd75536b","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:fsharp"],"facts":[{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"predicate":"presheaf_of_sets_on"},{"args":["urn:case:stacks:sh:fsharp","urn:case:stacks:sh:f"],"predicate":"plus_construction"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"universal_among_maps_to_sheaves","proof":true},"sources":[{"contentHash":"sha256:8504f30501fb40b30a60685a51aad76a416258206ec9a839891f7bf83cd38323","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006T","kind":"fragment","locator":"tag/006T","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:86546cd976c19579e8056b4fcbae7d781851f1b8799d12d0fba16321efaee59c","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_007Y","kind":"fragment","locator":"tag/007Y","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:282a15f0475c3f68a862ce8346d7972c4f62ab14daa09a1922c02bb9f01c558e","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-sheafification-sheaf}\nThe presheaf $\\mathcal{F}^{\\#}$ is a sheaf.\n\\end{lemma}"}]},{"contentHash":"sha256:9b004386054af3ebceab40302636bc759c8a1b7c732a3984765235c76fc000c4","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0080","kind":"fragment","locator":"tag/0080","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:2ba5dec16073f1fdafcf8a352bfc30f215d20caf060e768c51658875dc939220","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-sheafify-universal}\nLet $\\mathcal{F}$ be a presheaf of sets on $X$.\nAny map $\\mathcal{F} \\to \\mathcal{G}$ into a sheaf of sets\nfactors uniquely as\n$\\mathcal{F} \\to \\mathcal{F}^\\# \\to \\mathcal{G}$.\n\\end{lemma}"}]}],"text":"universal_among_maps_to_sheaves: **TRUE_ONLY** — установлено\nВыведено правом: sheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x); universal_among_maps_to_sheaves(urn:case:stacks:sh:f, urn:case:stacks:sh:fsharp)\nПрименены правила: SheafificationIsSheaf, SheafifyUniversal\nОтвет поражаем правилом «006T: a presheaf whose compatible families of sections do not glue is not a sheaf» — оно отменило бы вывод, будь установлено: compatible_sections_glue(urn:case:stacks:sh:fsharp); presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)\nОтвет поражаем правилом «006T: a presheaf in which a glued section is not unique is not a sheaf» — оно отменило бы вывод, будь установлено: gluing_is_unique(urn:case:stacks:sh:fsharp); presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)\nОтвет поражаем правилом «006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)» — оно отменило бы вывод, будь установлено: not_a_sheaf(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:6b62eb59e903…)\nproof-граф: 5 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["cohomology_universal_delta_functor(urn:case:stacks:sh:x)","abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx)","has_flasque_resolution(urn:case:stacks:sh:f)","has_injective_resolution(urn:case:stacks:sh:f)","sheaf_cohomology(urn:case:stacks:sh:x, 0, urn:case:stacks:sh:f, urn:case:stacks:sh:h0)"],"derivedOmitted":20,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"evidence":[],"id":"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"definition":{"concept":"urn:stacks:clir:categories#abelian_category","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"evidence":[],"id":"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372","kind":"rule_application","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-2"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"definition":{"concept":"urn:stacks:clir:categories#abelian_category","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"evidence":[],"id":"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","kind":"rule_application","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-3"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"evidence":[],"id":"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","kind":"rule_application","premises":["urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f"],"rule":"urn:stacks:clir:categories#LeftExactBySES","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#derived_functors_universal"},"evidence":[],"id":"urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02","kind":"rule_application","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor"},"evidence":[],"id":"urn:proof:apply:CohomologyUniversalDeltaFunctor:7ca182831cb1d0c9e1309d7ac0604a113df7a5f1c23a4f5f7217c114e488e89a","kind":"rule_application","premises":["urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"evidence":[],"id":"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","kind":"rule_application","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd"],"rule":"urn:stacks:clir:categories#LeftExactIsAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#rf_everywhere_defined"},"evidence":[],"id":"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6","kind":"rule_application","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#RFEverywhereDefined","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor"},"evidence":[],"id":"urn:proof:apply:DerivedFormDeltaFunctor:a0ef0823d7ce3dfbf6ae694696e8274c0bcf0f1aa97d2c7f6441b07934ca990e","kind":"rule_application","premises":["urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish"},"evidence":[],"id":"urn:proof:apply:NegativeDerivedVanish:7fa0ea40bfa7150a31985ea95de55d4a32279727237bc17d6ac1afeb8319f335","kind":"rule_application","premises":["urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#r0_agrees_with_f"},"evidence":[],"id":"urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b","kind":"rule_application","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-4"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-5"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"definition":{"concept":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"evidence":[],"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-6"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-7"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"evidence":[],"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"evidence":[],"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#object_of"},"evidence":[],"id":"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"evidence":[],"id":"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b"],"rule":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#has_injective_resolution"},"evidence":[],"id":"urn:proof:apply:InjectiveResolutionsExist:92a8097774445945984ed4264ea6947dc499382a6154bd9e66dd68af3f4abaaf","kind":"rule_application","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#InjectiveResolutionsExist","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-8"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-8","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":0},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology"},"evidence":[],"id":"urn:proof:apply:H0IsGlobalSections:f9d524c3bc12453d25b37f012849054d8fd6fb9113b84aff26549f16ad861424","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b","urn:proof:assert:urn:mcp:case#fact-3","urn:proof:assert:urn:mcp:case#fact-8"],"rule":"urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","kind":"constraint_check","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:categories#abelian_category/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585","kind":"constraint_check","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:categories#abelian_category/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749","kind":"constraint_check","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","kind":"constraint_check","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":0},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:H0IsGlobalSections:f9d524c3bc12453d25b37f012849054d8fd6fb9113b84aff26549f16ad861424"],"sourceAnchors":[]}],"proofHash":"sha256:1297155e73343af7a940705184b973b5c54d6b9003c74d353a94045985751f3b","roots":["urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585","urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749","urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","urn:proof:query:mcp"]},"resultHash":"sha256:8335998a5c3c0448bc102cbb5f729dd483db9fa0bb94e4279cea2385ade42067","schemaVersion":"law.core.evaluation/0.1"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006E","urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_01DZ","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_0109","urn:stacks:clir:categories#ST_010N"],"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"},{"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_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"}],"caseHash":"sha256:641768e375cd8c19e13427d6faf18341b5b8c48cad5ddfe40f742eff4f1279fb","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692","resultHash":"sha256:8335998a5c3c0448bc102cbb5f729dd483db9fa0bb94e4279cea2385ade42067","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"rulesApplied":["urn:stacks:clir:categories#LeftExactBySES","urn:stacks:clir:categories#LeftExactIsAdditive","urn:stacks:clir:categories#abelian_category/sufficient","urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","urn:stacks:clir:derived-functors#InjectiveResolutionsExist","urn:stacks:clir:derived-functors#NegativeDerivedVanish","urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","urn:stacks:clir:derived-functors#RFEverywhereDefined","urn:stacks:clir:sheaf-cohomology#AbAdditive","urn:stacks:clir:sheaf-cohomology#AbCoimageImage","urn:stacks:clir:sheaf-cohomology#AbCokernels","urn:stacks:clir:sheaf-cohomology#AbKernels","urn:stacks:clir:sheaf-cohomology#AbXAdditive","urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","urn:stacks:clir:sheaf-cohomology#AbXCokernels","urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","urn:stacks:clir:sheaf-cohomology#AbXKernels","urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor","urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections","urn:stacks:clir:sheaf-cohomology#SheafByGluing","urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient"],"signature":{"constants":{},"parameters":[{"labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}},{"labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Obj"}},{"labels":[],"name":"h","type":{"name":"urn:stacks:clir:category-theory#Obj"}}],"predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology","schemaVersion":"law.answers.signature/0.1","types":{"urn:law:std#Integer":{"kind":"std"},"urn:stacks:clir:category-theory#Obj":{"kind":"entity","labels":[{"language":"en","status":"official","text":"object of a category"},{"language":"ru","status":"unofficial","text":"объект категории"}],"namespace":"urn:stacks:clir:category-theory","package":"stacks.category_theory"},"urn:stacks:clir:sheaf-cohomology#Space":{"kind":"entity","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks.sheaf_cohomology"}},"vocab":{}},"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","missing":["not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"],"premises":[{"premise":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByRefutingCovering"}],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:641768e375cd8c19e13427d6faf18341b5b8c48cad5ddfe40f742eff4f1279fb\",\"decisionTime\":\"2026-09-06T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-09-06T12:00:00+05:00\",\"legalTime\":\"2026-09-06\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:5eba89ba52dadf1ac0a0af7c42fc79412b1ce5d7a2a433d2acef284612951d00\",\"semantics\":\"law.core/0.1.0\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaves_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#enough_injectives\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:categories#abelian_category\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#abelian_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\"],\"rule\":\"urn:stacks:clir:categories#abelian_category/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-2\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_groups_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-2\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:categories#abelian_category\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#abelian_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\"],\"rule\":\"urn:stacks:clir:categories#abelian_category/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-3\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#global_sections_functor\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-3\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#functor_between\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"},\"v3\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#preserves_left_exactness\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#left_exact_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\"],\"rule\":\"urn:stacks:clir:categories#LeftExactBySES\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#derived_functors_universal\"},\"evidence\":[],\"id\":\"urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"rule\":\"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:CohomologyUniversalDeltaFunctor:7ca182831cb1d0c9e1309d7ac0604a113df7a5f1c23a4f5f7217c114e488e89a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\"],\"rule\":\"urn:stacks:clir:categories#LeftExactIsAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#rf_everywhere_defined\"},\"evidence\":[],\"id\":\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"rule\":\"urn:stacks:clir:derived-functors#RFEverywhereDefined\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:DerivedFormDeltaFunctor:a0ef0823d7ce3dfbf6ae694696e8274c0bcf0f1aa97d2c7f6441b07934ca990e\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#negative_derived_functors_vanish\"},\"evidence\":[],\"id\":\"urn:proof:apply:NegativeDerivedVanish:7fa0ea40bfa7150a31985ea95de55d4a32279727237bc17d6ac1afeb8319f335\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#NegativeDerivedVanish\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#r0_agrees_with_f\"},\"evidence\":[],\"id\":\"urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-4\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-4\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-5\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_group_structure\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-5\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-6\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-6\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-7\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#gluing_is_unique\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-7\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafByGluing\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#object_of\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#has_injective_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:InjectiveResolutionsExist:92a8097774445945984ed4264ea6947dc499382a6154bd9e66dd68af3f4abaaf\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"rule\":\"urn:stacks:clir:derived-functors#InjectiveResolutionsExist\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-8\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:h0\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#global_sections\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-8\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":0},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:h0\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology\"},\"evidence\":[],\"id\":\"urn:proof:apply:H0IsGlobalSections:f9d524c3bc12453d25b37f012849054d8fd6fb9113b84aff26549f16ad861424\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b\",\"urn:proof:assert:urn:mcp:case#fact-3\",\"urn:proof:assert:urn:mcp:case#fact-8\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v3\":{\"id\":\"urn:case:stacks:sh:h0\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"requirementStatus\":\"NEITHER\",\"status\":\"UNDETERMINED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"literal\":{\"args\":[{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},{\"kind\":\"value\",\"type\":{\"name\":\"urn:law:std#Integer\"},\"value\":0},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:h0\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology\"},\"truthStatus\":\"TRUE_ONLY\"},\"evidence\":[],\"id\":\"urn:proof:query:mcp\",\"kind\":\"query_evaluation\",\"premises\":[\"urn:proof:apply:H0IsGlobalSections:f9d524c3bc12453d25b37f012849054d8fd6fb9113b84aff26549f16ad861424\"],\"sourceAnchors\":[]}],\"proofHash\":\"sha256:1297155e73343af7a940705184b973b5c54d6b9003c74d353a94045985751f3b\",\"roots\":[\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"urn:proof:query:mcp\"]},\"resultHash\":\"sha256:8335998a5c3c0448bc102cbb5f729dd483db9fa0bb94e4279cea2385ade42067\",\"results\":[{\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:mcp\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatusSupports\":[],\"proof\":\"urn:proof:query:mcp\",\"query\":\"urn:query:mcp\",\"resultKind\":\"PROPOSITION\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_category/necessary:8b36b74080734ae7\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_category/necessary:13c1fcb81161c70c\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:130d9bd8de768ef2\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[{\"description\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf: опора не установлена (§93.2)\",\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:130d9bd8de768ef2:input:0\",\"kind\":\"fact\",\"relatedNode\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf\"}],\"normativeStatus\":\"UNDETERMINED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"NEITHER\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_presheaf_on/necessary:58c8f7fa8bad2649\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.1\"}","evaluationSha256":"sha256:77517a1cb5de03a401c900ab80ad61a47a5b28fff69456e978b42f0a27aafb4d","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-2","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-3","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-4","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-5","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-6","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-7","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-8","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"}],"context":{"decisionTime":"2026-09-06T12:00:00+05:00","knowledgeTime":"2026-09-06T12:00:00+05:00","legalTime":"2026-09-06","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:8a58090dc9e51d0c78d83cb210fe43917cea71e0ee08f8f05532b76deb773e40","kind":"world_ref","nodeCount":383,"packages":[{"artifactHash":"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","semanticHash":"sha256:1ee2443e666f565715a3e3a5663914219d324c205aa6004f03c007d582687303"},{"artifactHash":"sha256:58778a4f746bcbfab9fdf432e1ba2b355fa02e4e0f2cefc1f41dec0f17f9cffc","namespace":"urn:stacks:clir:categories","package":"stacks-categories","semanticHash":"sha256:f21771fe43fa7adf28a188524efa50baebafb5892a4f79ebf8d717b9d3dcb1fc"},{"artifactHash":"sha256:99a2792dcc914bce6faddd3dca7875f19749b622e87c8e1f8a3007d8c6e6aaa3","namespace":"urn:stacks:clir:category-theory","package":"stacks-category-theory","semanticHash":"sha256:bd6536b4675c408f93bc08df24740211450e93a567c136fbad33194dfe200cc2"},{"artifactHash":"sha256:c522aaa206f22044b085107a030f77422be0f9478c88125c82a8f3b9ade621e3","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","semanticHash":"sha256:c699babe3fee3095884ef65f76edb67bf4c7c42c15f46f4e75ab10bc248922d7"}],"programHash":"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":0},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.1.0"},"requestCanonicalSha256":"sha256:6232dc07bee5955f948dc9b93e6b463aa1e8e580f79ce1da221ac070e6773135"},"id":"h0","kind":"truth","label":"H⁰ равна глобальным сечениям","presentation":{"blockers":{"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","premises":[{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"missing":true,"predicate":"not_a_sheaf","predicateId":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","status":"NEITHER","text":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"}],"rule":"NotSheafByRefutingCovering","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering"}]},"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:6232dc07bee5955f948dc9b93e6b463aa1e8e580f79ce1da221ac070e6773135"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"id":"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"id":"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"id":"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"id":"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01DG"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"id":"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXKernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"id":"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":["urn:stacks:clir:categories#ST_0109"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Category"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"id":"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbAdditive","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"id":"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"id":"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCokernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"id":"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbKernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"id":"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":["urn:stacks:clir:categories#ST_0109"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Category"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"},{"kind":"var","var":"v2"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"id":"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0716"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Functor"}},{"id":"v1","type":{"name":"stacks.category_theory#Category"}},{"id":"v2","type":{"name":"stacks.category_theory#Category"}},{"id":"v3","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"id":"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0716"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"id":"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f"],"rule":"urn:stacks:clir:categories#LeftExactBySES","sourceAnchors":["urn:stacks:clir:categories#ST_010N"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Functor"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"id":"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd"],"rule":"urn:stacks:clir:categories#LeftExactIsAdditive","sourceAnchors":["urn:stacks:clir:categories#ST_010N"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Functor"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#rf_everywhere_defined"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#rf_everywhere_defined"},"id":"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#RFEverywhereDefined","sourceAnchors":["urn:stacks:clir:derived-functors#ST_05TI","urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:derived-functors#Category"}},{"id":"v2","type":{"name":"urn:stacks:clir:derived-functors#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#r0_agrees_with_f"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#r0_agrees_with_f"},"id":"urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","sourceAnchors":["urn:stacks:clir:derived-functors#ST_05TD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:derived-functors#Functor"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006K"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0070"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections"},"id":"urn:proof:assert:urn:mcp:case#fact-8","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":0},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology"},"head":{"args":[{"kind":"var","var":"v1"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":0},{"kind":"var","var":"v2"},{"kind":"var","var":"v3"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology"},"id":"urn:proof:apply:H0IsGlobalSections:f9d524c3bc12453d25b37f012849054d8fd6fb9113b84aff26549f16ad861424","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b","urn:proof:assert:urn:mcp:case#fact-3","urn:proof:assert:urn:mcp:case#fact-8"],"rule":"urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006E","urn:stacks:clir:sheaf-cohomology#ST_0716"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"id":"v2","type":{"name":"stacks.category_theory#Obj"}},{"id":"v3","type":{"name":"stacks.category_theory#Obj"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"kind":"value","type":{"name":"urn:law:std#Integer"},"value":0},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:h0","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:H0IsGlobalSections:f9d524c3bc12453d25b37f012849054d8fd6fb9113b84aff26549f16ad861424"],"trust":"engine"}],"symbols":[{"id":"urn:stacks:clir:categories#LeftExactBySES","kind":"rule","labels":[{"language":"en","status":"official","text":"010N (2): \\(F\\) is left exact if 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":"010N (2): \\(F\\) точен слева, если для всякой короткой точной последовательности \\(0 → A → B → C → 0\\) последовательность \\(0 → F(A) → F(B) → F(C)\\) точна"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#LeftExactIsAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"010N (1): if \\(F\\) is left exact, then it is additive"},{"language":"ru","status":"unofficial","text":"010N (1): если \\(F\\) точен слева, то он аддитивен"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#abelian_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"name":"abelian_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#abelian_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#abelian_category/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"package":"urn:stacks:clir:categories"},{"id":"urn:stacks:clir:categories#abelian_category/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#additive_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0104: the category is additive"},{"language":"ru","status":"unofficial","text":"0104: категория аддитивна"}],"name":"additive_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#additive_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#additive_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is additive"},{"language":"ru","status":"unofficial","text":"функтор аддитивен"}],"name":"additive_functor","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#additive_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:categories#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#coimage_to_image_isomorphism","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the natural map \\(Coim(f) → Im(f)\\) is an isomorphism for all morphisms \\(f\\) of the 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#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#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#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#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:sheaf-cohomology#AbAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism for every morphism"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм для всякого морфизма"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","kind":"rule","labels":[{"language":"en","status":"official","text":"01DG: the category of abelian sheaves on a topological space \\(X\\) has enough injectives"},{"language":"ru","status":"unofficial","text":"01DG: в категории абелевых пучков на топологическом пространстве \\(X\\) достаточно инъективных"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (1): an abelian presheaf on \\(X\\) whose underlying presheaf of sets is a sheaf is an abelian sheaf"},{"language":"ru","status":"unofficial","text":"0070 (1): абелев предпучок на \\(X\\), чей предпучок множеств — пучок, есть абелев пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (2): an abelian sheaf on \\(X\\) is an object of \\(Ab(X)\\)"},{"language":"ru","status":"unofficial","text":"0070 (2): абелев пучок на \\(X\\) — объект категории \\(Ab(X)\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"01DZ: the functors \\(H^i(X, −)\\) form a universal \\(δ\\)-functor \\(Ab(X) → Ab\\)"},{"language":"ru","status":"unofficial","text":"01DZ: функторы \\(H^i(X, −)\\) образуют универсальный \\(δ\\)-функтор \\(Ab(X) → Ab\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a functor from \\(Ab(X)\\) to \\(Ab\\)"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — функтор из \\(Ab(X)\\) в \\(Ab\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a left exact functor"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — точный слева функтор"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","kind":"rule","labels":[{"language":"en","status":"official","text":"0FKS with 01AD: every abelian sheaf \\(F\\) on \\(X\\) has the Godement resolution \\(0 → F → f_*f^*F → …\\) by flasque sheaves"},{"language":"ru","status":"unofficial","text":"0FKS с 01AD: у всякого абелева пучка \\(F\\) на \\(X\\) есть резольвента Годемана \\(0 → F → f_*f^*F → …\\) вялыми пучками"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections","kind":"rule","labels":[{"language":"en","status":"official","text":"006E with 05TD (3): \\(H^0(X, F) = Γ(X, F)\\) since \\(Γ(X, −)\\) is left exact"},{"language":"ru","status":"unofficial","text":"006E с 05TD (3): \\(H^0(X, F) = Γ(X, F)\\), поскольку \\(Γ(X, −)\\) точен слева"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"006T: предпучок, опровергнутый открытым покрытием, — не пучок множеств на \\(X\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section"},{"language":"ru","status":"unofficial","text":"006T: предпучок множеств — пучок, если согласованные семейства сечений над любым открытым покрытием склеиваются в единственное сечение"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"each \\(F(U)\\) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups"},{"language":"ru","status":"unofficial","text":"каждое \\(F(U)\\) несёт структуру абелевой группы, и все отображения ограничения — гомоморфизмы абелевых групп"}],"name":"abelian_group_structure","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab\\), the category of abelian groups"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab\\) — категория абелевых групп"}],"name":"abelian_groups_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category/arg/ab","labels":[],"name":"ab","type":{"name":"stacks.category_theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"name":"abelian_presheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0070: \\(F\\) is an abelian sheaf on \\(X\\) — an abelian presheaf whose underlying presheaf of sets is a sheaf"},{"language":"ru","status":"unofficial","text":"0070: \\(F\\) — абелев пучок на \\(X\\): абелев предпучок, чей предпучок множеств — пучок"}],"name":"abelian_sheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01DZ: the family \\(H^i(X, −)\\) forms a universal \\(δ\\)-functor from \\(Ab(X)\\) to \\(Ab\\)"},{"language":"ru","status":"unofficial","text":"01DZ: семейство \\(H^i(X, −)\\) образует универсальный \\(δ\\)-функтор из \\(Ab(X)\\) в \\(Ab\\)"}],"name":"cohomology_universal_delta_functor","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, existence: for every open covering \\(U = ∪ U_i\\) and sections \\(s_i ∈ F(U_i)\\) with \\(s_i|U_i∩U_j = s_j|U_i∩U_j\\) there exists \\(s ∈ F(U)\\) with \\(s|U_i = s_i\\)"},{"language":"ru","status":"unofficial","text":"006T, существование: для всякого открытого покрытия \\(U = ∪ U_i\\) и сечений \\(s_i ∈ F(U_i)\\), согласованных на пересечениях, существует \\(s ∈ F(U)\\) с \\(s|U_i = s_i\\)"}],"name":"compatible_sections_glue","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(Γ(X, F) = F(X)\\) is the given abelian group"},{"language":"ru","status":"unofficial","text":"\\(Γ(X, F) = F(X)\\) — данная абелева группа"}],"name":"global_sections","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#global_sections/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections/arg/h","labels":[],"name":"h","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"},{"language":"ru","status":"unofficial","text":"функтор есть \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"}],"name":"global_sections_functor","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Functor"}},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, uniqueness: a section \\(s ∈ F(U)\\) is determined by its restrictions \\(s|U_i\\) to an open covering"},{"language":"ru","status":"unofficial","text":"006T, единственность: сечение \\(s ∈ F(U)\\) определяется своими ограничениями \\(s|U_i\\) на открытое покрытие"}],"name":"gluing_is_unique","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)"},{"language":"ru","status":"unofficial","text":"0FKS: у пучка есть функториальная резольвента вялыми пучками (резольвента Годемана)"}],"name":"has_flasque_resolution","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is not a sheaf on \\(X\\): the sheaf condition 006T fails on some open covering"},{"language":"ru","status":"unofficial","text":"\\(F\\) — не пучок на \\(X\\): условие 006T провалено на некотором открытом покрытии"}],"name":"not_a_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a presheaf of sets on \\(X\\): a rule assigning a set \\(F(U)\\) to each open \\(U\\) and restriction maps \\(ρ^U_V\\) to inclusions \\(V ⊂ U\\) with \\(ρ^U_U = id\\) and \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — предпучок множеств на \\(X\\): правило, сопоставляющее каждому открытому \\(U\\) множество \\(F(U)\\), а вложениям \\(V ⊂ U\\) — отображения ограничения \\(ρ^U_V\\) с \\(ρ^U_U = id\\) и \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"}],"name":"presheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01DZ: \\(H^n(X, F)\\), the n-th cohomology group of the abelian sheaf \\(F\\), is the given object"},{"language":"ru","status":"unofficial","text":"01DZ: \\(H^n(X, F)\\), n-я группа когомологий абелева пучка \\(F\\), — данный объект"}],"name":"sheaf_cohomology","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology/arg/n","labels":[],"name":"n","type":{"name":"urn:law:std#Integer"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_cohomology/arg/h","labels":[],"name":"h","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — пучок множеств на \\(X\\)"}],"name":"sheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab(X)\\), the category of sheaves of abelian groups on \\(X\\)"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab(X)\\) — категория пучков абелевых групп на \\(X\\)"}],"name":"sheaves_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/c","labels":[],"name":"c","type":{"name":"stacks.category_theory#Category"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006E","urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_01DZ","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_0109","urn:stacks:clir:categories#ST_010N"],"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"},{"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_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"}],"caseHash":"sha256:641768e375cd8c19e13427d6faf18341b5b8c48cad5ddfe40f742eff4f1279fb","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692","resultHash":"sha256:8335998a5c3c0448bc102cbb5f729dd483db9fa0bb94e4279cea2385ade42067","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:sh:x",0,"urn:case:stacks:sh:f","urn:case:stacks:sh:h0"],"facts":[{"args":["urn:case:stacks:sh:abx","urn:case:stacks:sh:x"],"predicate":"sheaves_category"},{"args":["urn:case:stacks:sh:ab"],"predicate":"abelian_groups_category"},{"args":["urn:case:stacks:sh:gamma","urn:case:stacks:sh:x"],"predicate":"global_sections_functor"},{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"predicate":"presheaf_of_sets_on"},{"args":["urn:case:stacks:sh:f"],"predicate":"abelian_group_structure"},{"args":["urn:case:stacks:sh:f"],"predicate":"compatible_sections_glue"},{"args":["urn:case:stacks:sh:f"],"predicate":"gluing_is_unique"},{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x","urn:case:stacks:sh:h0"],"predicate":"global_sections"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"sheaf_cohomology","proof":true},"sources":[{"contentHash":"sha256:5940ad0150140fcc429bfcb66cf91e0303638f57e88332528ed64e0204a1cc00","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006E","kind":"fragment","locator":"tag/006E","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e728b50fec00b5a51ffd85b7518d87d42e03d9b363884124cfca05a3aa077563","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-presheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf $\\mathcal{F}$ of sets on $X$} is a rule which\nassigns to each open $U \\subset X$ a set $\\mathcal{F}(U)$ and\nto each inclusion $V \\subset U$ a map\n$\\rho^U_V : \\mathcal{F}(U) \\to \\mathcal{F}(V)$ such that\n$\\rho^U_U = \\text{id}_{\\mathcal{F}(U)}$ and\nwhenever $W \\subset V \\subset U$ we have\n$\\rho^U_W = \\rho^V_W \\circ \\rho ^U_V$.\n\\item A {\\it morphism $\\varphi : \\mathcal{F} \\to \\mathcal{G}$\nof presheaves of sets on $X$} is a rule which assigns to each\nopen $U \\subset X$ a map of sets $\\varphi : \\mathcal{F}(U)\n\\to \\mathcal{G}(U)$ compatible with restriction maps,\ni.e., whenever $V \\subset U \\subset X$ are open the\ndiagram\n$$\n\\xymatrix{\n\\mathcal{F}(U) \\ar[r]^\\varphi \\ar[d]^{\\rho^U_V} &\n\\mathcal{G}(U) \\ar[d]^{\\rho^U_V} \\\\\n\\mathcal{F}(V) \\ar[r]^\\varphi & \\mathcal{G}(V)\n}\n$$\ncommutes.\n\\item The category of presheaves of sets on $X$ will be denoted\n$\\textit{PSh}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:fa7c3dec36920848465ed2902237f8efade0877767eebcee37676bb1bb8717d9","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006K","kind":"fragment","locator":"tag/006K","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:b8388b54fb29de485ebd7c91f38c607f6005d41b4d427265a75b163118fd70d6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-presheaves}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf of abelian groups on $X$} or an\n{\\it abelian presheaf over $X$}\nis a presheaf of sets $\\mathcal{F}$ such that for each open\n$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with\nthe structure of an abelian group, and such that all restriction\nmaps $\\rho^U_V$ are homomorphisms of abelian groups, see\nLemma \\ref{lemma-abelian-presheaves} above.\n\\item A {\\it morphism of abelian presheaves over $X$}\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves\nof sets which induces\na homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nfor every open $U \\subset X$.\n\\item The category of presheaves of abelian groups on $X$ is denoted\n$\\textit{PAb}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:8504f30501fb40b30a60685a51aad76a416258206ec9a839891f7bf83cd38323","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006T","kind":"fragment","locator":"tag/006T","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:04f51a87121abeee45cd4e99f1379db299b359b4ac801017c7770a7f05d2c370","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_0070","kind":"fragment","locator":"tag/0070","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3260f7d5e9cbfe266160539b8375cd23ac2429762aaa12eb860ad707dc28a6cd","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item An {\\it abelian sheaf on $X$} or\n{\\it sheaf of abelian groups on $X$}\nis an abelian presheaf on $X$ such that the underlying presheaf of\nsets is a sheaf.\n\\item The category of sheaves of abelian groups\nis denoted $\\textit{Ab}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:d0262bf7656850a1933eef167a360437aa2bb6436e57461a3d23de09e8cc8ef3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01AD","kind":"fragment","locator":"tag/01AD","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf","language":"en","status":"official","text":"\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we work out basic notions of sheaves of modules.\nThis in particular includes the case of abelian sheaves, since\nthese may be viewed as sheaves of $\\underline{\\mathbf{Z}}$-modules.\nBasic references are \\cite{FAC}, \\cite{EGA} and \\cite{SGA4}.\n\n\\medskip\\noindent\nWe work out what happens for sheaves of modules on ringed topoi\nin another chapter (see\nModules on Sites, Section \\ref{sites-modules-section-introduction}),\nalthough there we will mostly just duplicate the discussion\nfrom this chapter."}]},{"contentHash":"sha256:1bf6bd6db42698470b675022b39392436590d133c6e4e0e6f342e63ce50e5290","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01AG","kind":"fragment","locator":"tag/01AG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:d40ca7777a52e10948b9c7e8da68432632fc0b8f4bb33c427370acaf48a4b19f","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover\na complex\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nis exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the\ncomplex\n$$\n\\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x\n$$\nis exact at $\\mathcal{G}_x$.\n\\end{lemma}"}]},{"contentHash":"sha256:48de83780b89577a36cf370ecf47370fa86fe2d17276908d3fb411375d5e1ec7","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_INJECTIVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01DG","kind":"fragment","locator":"tag/01DG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e4b4f09acb0a39efab590f7a0a1974dde000a27f6205d5284eb9b41b91b7f6d6","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian-sheaves-space}\nLet $X$ be a topological space.\nThe category of abelian sheaves on $X$ has enough injectives.\nIn fact it has functorial injective embeddings.\n\\end{lemma}"}]},{"contentHash":"sha256:b1dd6e78bfe8178dd35b60b16af0e740bb77f76f64096d01aaa44a811d4491b3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01DZ","kind":"fragment","locator":"tag/01DZ","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:992dfb482cc74c054eb720fd53a56dd386827983f1cb4214bf3e7230a56ea9d3","language":"en","status":"official","text":"\\section{Cohomology of sheaves}\n\\label{section-cohomology-sheaves}\n\n\\noindent\nLet $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf.\nWe know that the category of abelian sheaves on $X$ has enough injectives, see\nInjectives, Lemma \\ref{injectives-lemma-abelian-sheaves-space}.\nHence we can choose an injective resolution\n$\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$. As is customary we define\n\\begin{equation}\n\\label{equation-cohomology}\nH^i(X, \\mathcal{F}) = H^i(\\Gamma(X, \\mathcal{I}^\\bullet))\n\\end{equation}\nto be the {\\it $i$th cohomology group of the abelian sheaf $\\mathcal{F}$}.\nThe family of functors $H^i(X, -)$ forms a universal $\\delta$-functor\nfrom $\\textit{Ab}(X) \\to \\textit{Ab}$.\n\n\\medskip\\noindent\nLet $f : X \\to Y$ be a continuous map of topological spaces. With\n$\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$ as above\nwe define\n\\begin{equation}\n\\label{equation-higher-direct-image}\nR^if_*\\mathcal{F} = H^i(f_*\\mathcal{I}^\\bullet)\n\\end{equation}\nto be the {\\it $i$th higher direct image of $\\mathcal{F}$}.\nThe family of functors $R^if_*$ forms a universal $\\delta$-functor\nfrom $\\textit{Ab}(X) \\to \\textit{Ab}(Y)$.\n\n\\medskip\\noindent\nLet $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an\n$\\mathcal{O}_X$-module. We know that the category of $\\mathcal{O}_X$-modules\non $X$ has enough injectives, see\nInjectives, Lemma \\ref{injectives-lemma-sheaves-modules-space}.\nHence we can choose an injective resolution\n$\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$. As is customary we define\n\\begin{equation}\n\\label{equation-cohomology-modules}\nH^i(X, \\mathcal{F}) = H^i(\\Gamma(X, \\mathcal{I}^\\bullet))\n\\end{equation}\nto be the {\\it $i$th cohomology group of $\\mathcal{F}$}.\nThe family of functors $H^i(X, -)$ forms a universal $\\delta$-functor\nfrom $\\textit{Mod}(\\mathcal{O}_X) \\to \\text{Mod}_{\\mathcal{O}_X(X)}$.\n\n\\medskip\\noindent\nLet $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. With $\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$ as above\nwe define\n\\begin{equation}\n\\label{equation-higher-direct-image-modules}\nR^if_*\\mathcal{F} = H^i(f_*\\mathcal{I}^\\bullet)\n\\end{equation}\nto be the {\\it $i$th higher direct image of $\\mathcal{F}$}.\nThe family of functors $R^if_*$ forms a universal $\\delta$-functor\nfrom $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$."}]},{"contentHash":"sha256:0eee6ff9b997a29e65a9204cd10ad0d6c3337360eac630526713e3e18f1c9b03","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_0716","kind":"fragment","locator":"tag/0716","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:f61344b89b5447a2349b548070137ca4e6f491c90033bf637527fbd9f371c876","language":"en","status":"official","text":"\\section{Derived functors}\n\\label{section-derived-functors}\n\n\\noindent\nWe briefly explain how to get right derived functors using resolution\nfunctors. For the unbounded derived functors, please see\nSection \\ref{section-unbounded}.\n\n\\medskip\\noindent\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is abelian, see\nModules, Lemma \\ref{modules-lemma-abelian}.\nIn this chapter we will write\n$$\nK(\\mathcal{O}_X) = K(\\textit{Mod}(\\mathcal{O}_X))\n\\quad\n\\text{and}\n\\quad\nD(\\mathcal{O}_X) = D(\\textit{Mod}(\\mathcal{O}_X)).\n$$\nand similarly for the bounded versions for the triangulated categories\nintroduced in\nDerived Categories, Definition \\ref{derived-definition-complexes-notation} and\nDefinition \\ref{derived-definition-unbounded-derived-category}.\nBy\nDerived Categories, Remark \\ref{derived-remark-big-abelian-category}\nthere exists a resolution functor\n$$\nj = j_X :\nK^{+}(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK^{+}(\\mathcal{I})\n$$\nwhere $\\mathcal{I}$ is the strictly full additive subcategory of\n$\\textit{Mod}(\\mathcal{O}_X)$ consisting of injective sheaves.\nFor any left exact functor\n$F : \\textit{Mod}(\\mathcal{O}_X) \\to \\mathcal{B}$\ninto any abelian category $\\mathcal{B}$ we will denote $RF$ the\nright derived functor described in\nDerived Categories, Section \\ref{derived-section-right-derived-functor}\nand constructed using the resolution functor $j_X$ just described:\n\\begin{equation}\n\\label{equation-RF}\nRF = F \\circ j_X' : D^{+}(X) \\longrightarrow D^{+}(\\mathcal{B})\n\\end{equation}\nsee\nDerived Categories, Lemma \\ref{derived-lemma-right-derived-functor}\nfor notation. Note that we may think of $RF$ as defined on\n$\\textit{Mod}(\\mathcal{O}_X)$,\n$\\text{Comp}^{+}(\\textit{Mod}(\\mathcal{O}_X))$,\n$K^{+}(X)$, or $D^{+}(X)$\ndepending on the situation. According to\nDerived Categories, Definition \\ref{derived-definition-higher-derived-functors}\nwe obtain the $i$th right derived functor\n\\begin{equation}\n\\label{equation-RFi}\nR^iF = H^i \\circ RF : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\mathcal{B}\n\\end{equation}\nso that $R^0F = F$ and $\\{R^iF, \\delta\\}_{i \\geq 0}$ is universal\n$\\delta$-functor, see\nDerived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}.\n\n\\medskip\\noindent\nHere are two special cases of this construction.\nGiven a ring $R$ we write $K(R) = K(\\text{Mod}_R)$ and\n$D(R) = D(\\text{Mod}_R)$ and similarly for bounded versions.\nFor any open $U \\subset X$ we have a left exact functor\n$\n\\Gamma(U, -) :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\text{Mod}_{\\mathcal{O}_X(U)}\n$\nwhich gives rise to\n\\begin{equation}\n\\label{equation-total-derived-cohomology}\nR\\Gamma(U, -) :\nD^{+}(X)\n\\longrightarrow\nD^{+}(\\mathcal{O}_X(U))\n\\end{equation}\nby the discussion above. We set $H^i(U, -) = R^i\\Gamma(U, -)$.\nIf $U = X$ we recover (\\ref{equation-cohomology-modules}).\nIf $f : X \\to Y$ is a morphism of ringed spaces, then we have\nthe left exact functor\n$\nf_* :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_Y)\n$\nwhich gives rise to the {\\it derived pushforward}\n\\begin{equation}\n\\label{equation-total-derived-direct-image}\nRf_* :\nD^{+}(X)\n\\longrightarrow\nD^{+}(Y)\n\\end{equation}\nThe $i$th cohomology sheaf of $Rf_*\\mathcal{F}^\\bullet$ is denoted\n$R^if_*\\mathcal{F}^\\bullet$ and called the $i$th {\\it higher direct image}\nin accordance with (\\ref{equation-higher-direct-image-modules}).\nThe two displayed functors above are exact functors\nof derived categories.\n\n\\medskip\\noindent\n{\\bf Abuse of notation:} When the functor $Rf_*$, or any other\nderived functor, is applied to a sheaf $\\mathcal{F}$ on $X$ or a complex\nof sheaves it is understood that $\\mathcal{F}$ has been replaced by a\nsuitable resolution of $\\mathcal{F}$. To facilitate this kind of\noperation we will say, given an object\n$\\mathcal{F}^\\bullet \\in D(\\mathcal{O}_X)$,\nthat a bounded below complex $\\mathcal{I}^\\bullet$ of injectives of\n$\\textit{Mod}(\\mathcal{O}_X)$\n{\\it represents $\\mathcal{F}^\\bullet$ in the derived category}\nif there exists a quasi-isomorphism\n$\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$. In the same vein the phrase\n``let $\\alpha : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be\na morphism of $D(\\mathcal{O}_X)$''\ndoes not mean that $\\alpha$ is represented by a\nmorphism of complexes. If we have an actual morphism of complexes we will\nsay so."}]},{"contentHash":"sha256:787503a0ec46db68e4e11e9bf5fa2fabae3304c806bc7e2f5be69ab1af6cc11f","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0FKS","kind":"fragment","locator":"tag/0FKS","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:8b044713268f8366ca6cc0212d88985ff0517eb2417f464d218f454e93c283fe","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-godement-resolution}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution\n$$\n0 \\to\n\\mathcal{F} \\to\nf_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots\n$$\nfunctorial in $\\mathcal{F}$ such that each term\n$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque\n$\\mathcal{O}_X$-module and such that for all $x \\in X$ the\nmap\n$$\n\\mathcal{F}_x[0] \\to \\Big(\n(f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots\n\\Big)\n$$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.\n\\end{lemma}"}]},{"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":"sha256:9698ad04b03a0ca888a87e8f88e9925a856764046ae30cfb569730f33de5c92d","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":"sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be","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}"}]}],"text":"sheaf_cohomology: **TRUE_ONLY** — установлено\nВыведено правом: cohomology_universal_delta_functor(urn:case:stacks:sh:x); abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx); has_flasque_resolution(urn:case:stacks:sh:f); has_injective_resolution(urn:case:stacks:sh:f); sheaf_cohomology(urn:case:stacks:sh:x, 0, urn:case:stacks:sh:f, urn:case:stacks:sh:h0)\n…и ещё 20 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, InjectiveResolutionsExist, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined, AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXEnoughInjectives, AbXKernels, AbelianSheafByDefinition, AbelianSheafIsObjectOfAbX, CohomologyUniversalDeltaFunctor, GlobalSectionsBetween, GlobalSectionsLeftExact, GodementResolutionExists, H0IsGlobalSections, SheafByGluing, abelian_presheaf_on/sufficient\nОтвет поражаем правилом «006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)» — оно отменило бы вывод, будь установлено: not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:037863663edc…)\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; Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина\nproof-граф: 41 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["cohomology_universal_delta_functor(urn:case:stacks:sh:x)","abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx)","has_flasque_resolution(urn:case:stacks:sh:f)","has_injective_resolution(urn:case:stacks:sh:f)","flasque(urn:case:stacks:sh:f)","right_acyclic_for(urn:case:stacks:sh:f, urn:case:stacks:sh:gamma)","higher_derived_vanish(urn:case:stacks:sh:gamma, urn:case:stacks:sh:f)","cohomology_vanishes_positive(urn:case:stacks:sh:x, urn:case:stacks:sh:f)"],"derivedOmitted":20,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"evidence":[],"id":"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"definition":{"concept":"urn:stacks:clir:categories#abelian_category","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"evidence":[],"id":"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372","kind":"rule_application","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-2"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"definition":{"concept":"urn:stacks:clir:categories#abelian_category","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"evidence":[],"id":"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","kind":"rule_application","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-3"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"evidence":[],"id":"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","kind":"rule_application","premises":["urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f"],"rule":"urn:stacks:clir:categories#LeftExactBySES","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#derived_functors_universal"},"evidence":[],"id":"urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02","kind":"rule_application","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor"},"evidence":[],"id":"urn:proof:apply:CohomologyUniversalDeltaFunctor:7ca182831cb1d0c9e1309d7ac0604a113df7a5f1c23a4f5f7217c114e488e89a","kind":"rule_application","premises":["urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"evidence":[],"id":"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","kind":"rule_application","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd"],"rule":"urn:stacks:clir:categories#LeftExactIsAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#rf_everywhere_defined"},"evidence":[],"id":"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6","kind":"rule_application","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#RFEverywhereDefined","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor"},"evidence":[],"id":"urn:proof:apply:DerivedFormDeltaFunctor:a0ef0823d7ce3dfbf6ae694696e8274c0bcf0f1aa97d2c7f6441b07934ca990e","kind":"rule_application","premises":["urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#negative_derived_functors_vanish"},"evidence":[],"id":"urn:proof:apply:NegativeDerivedVanish:7fa0ea40bfa7150a31985ea95de55d4a32279727237bc17d6ac1afeb8319f335","kind":"rule_application","premises":["urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#NegativeDerivedVanish","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#r0_agrees_with_f"},"evidence":[],"id":"urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b","kind":"rule_application","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-4"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-5"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"definition":{"concept":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"evidence":[],"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-6"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-7"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"evidence":[],"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"evidence":[],"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#object_of"},"evidence":[],"id":"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"evidence":[],"id":"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b"],"rule":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#has_injective_resolution"},"evidence":[],"id":"urn:proof:apply:InjectiveResolutionsExist:92a8097774445945984ed4264ea6947dc499382a6154bd9e66dd68af3f4abaaf","kind":"rule_application","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#InjectiveResolutionsExist","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-8"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#restrictions_surjective"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-8","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"evidence":[],"id":"urn:proof:apply:FlasqueByRestrictions:34f730ec0a1b8af2d0ae379c5a75880216a66b9739f6f1d5d50b2c80867dff8f","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-8"],"rule":"urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#right_acyclic_for"},"evidence":[],"id":"urn:proof:apply:FlasqueIsAcyclic:eaa95f4954a00a93fffd4cad8ae80bb14882236dcfa2fc0ad1c73116fabe86e1","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:apply:FlasqueByRestrictions:34f730ec0a1b8af2d0ae379c5a75880216a66b9739f6f1d5d50b2c80867dff8f","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#higher_derived_vanish"},"evidence":[],"id":"urn:proof:apply:HigherDerivedVanishForAcyclic:6d47058a5fc5f3ad5543f1092b3393395165132d7b4775dfb2e8b828bea19ee8","kind":"rule_application","premises":["urn:proof:apply:FlasqueIsAcyclic:eaa95f4954a00a93fffd4cad8ae80bb14882236dcfa2fc0ad1c73116fabe86e1","urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive"},"evidence":[],"id":"urn:proof:apply:CohomologyVanishesForAcyclic:a47cde96682458b0dac7ead3e82920921c7e4df46c52e53c52f25a310e17e8ce","kind":"rule_application","premises":["urn:proof:apply:HigherDerivedVanishForAcyclic:6d47058a5fc5f3ad5543f1092b3393395165132d7b4775dfb2e8b828bea19ee8","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","kind":"constraint_check","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:categories#abelian_category/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585","kind":"constraint_check","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:categories#abelian_category/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749","kind":"constraint_check","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","kind":"constraint_check","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:CohomologyVanishesForAcyclic:a47cde96682458b0dac7ead3e82920921c7e4df46c52e53c52f25a310e17e8ce"],"sourceAnchors":[]}],"proofHash":"sha256:5d912b4bf938e63066eea6b355a3ab60755c59d0dce91da465e7f8033958330d","roots":["urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585","urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749","urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","urn:proof:query:mcp"]},"resultHash":"sha256:1166b2901cc0036291a26774c3e5724613084e2ab2033e1d286dca27e0aa25d3","schemaVersion":"law.core.evaluation/0.1"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_01DZ","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_09SW","urn:stacks:clir:sheaf-cohomology#ST_09SY","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_0109","urn:stacks:clir:categories#ST_010N"],"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"},{"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_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"}],"caseHash":"sha256:eb097105acd6264efc4b3a976119c21291db893e9fd17e38d38569e961f8c9fa","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692","resultHash":"sha256:1166b2901cc0036291a26774c3e5724613084e2ab2033e1d286dca27e0aa25d3","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"rulesApplied":["urn:stacks:clir:categories#LeftExactBySES","urn:stacks:clir:categories#LeftExactIsAdditive","urn:stacks:clir:categories#abelian_category/sufficient","urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor","urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor","urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic","urn:stacks:clir:derived-functors#InjectiveResolutionsExist","urn:stacks:clir:derived-functors#NegativeDerivedVanish","urn:stacks:clir:derived-functors#R0AgreesIfLeftExact","urn:stacks:clir:derived-functors#RFEverywhereDefined","urn:stacks:clir:sheaf-cohomology#AbAdditive","urn:stacks:clir:sheaf-cohomology#AbCoimageImage","urn:stacks:clir:sheaf-cohomology#AbCokernels","urn:stacks:clir:sheaf-cohomology#AbKernels","urn:stacks:clir:sheaf-cohomology#AbXAdditive","urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","urn:stacks:clir:sheaf-cohomology#AbXCokernels","urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","urn:stacks:clir:sheaf-cohomology#AbXKernels","urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor","urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic","urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions","urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","urn:stacks:clir:sheaf-cohomology#SheafByGluing","urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient"],"signature":{"constants":{},"parameters":[{"labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Obj"}}],"predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive","schemaVersion":"law.answers.signature/0.1","types":{"urn:stacks:clir:category-theory#Obj":{"kind":"entity","labels":[{"language":"en","status":"official","text":"object of a category"},{"language":"ru","status":"unofficial","text":"объект категории"}],"namespace":"urn:stacks:clir:category-theory","package":"stacks.category_theory"},"urn:stacks:clir:sheaf-cohomology#Space":{"kind":"entity","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks.sheaf_cohomology"}},"vocab":{}},"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","missing":["not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"],"premises":[{"premise":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByRefutingCovering"}],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:eb097105acd6264efc4b3a976119c21291db893e9fd17e38d38569e961f8c9fa\",\"decisionTime\":\"2026-09-06T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-09-06T12:00:00+05:00\",\"legalTime\":\"2026-09-06\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:570cf6ebc4a344b69178afa352de387bec17c0efe047326f0f34674677fc0baa\",\"semantics\":\"law.core/0.1.0\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaves_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#enough_injectives\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:categories#abelian_category\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#abelian_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\"],\"rule\":\"urn:stacks:clir:categories#abelian_category/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-2\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_groups_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-2\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:categories#abelian_category\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#abelian_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\"],\"rule\":\"urn:stacks:clir:categories#abelian_category/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-3\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#global_sections_functor\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-3\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#functor_between\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"},\"v3\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#preserves_left_exactness\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#left_exact_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\"],\"rule\":\"urn:stacks:clir:categories#LeftExactBySES\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#derived_functors_universal\"},\"evidence\":[],\"id\":\"urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"rule\":\"urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:CohomologyUniversalDeltaFunctor:7ca182831cb1d0c9e1309d7ac0604a113df7a5f1c23a4f5f7217c114e488e89a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:DerivedUniversalDeltaFunctor:d90d3681b6d0881591567b91e6b77945bb216a1f4b384f7b75435c67e3271a02\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\"],\"rule\":\"urn:stacks:clir:categories#LeftExactIsAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#rf_everywhere_defined\"},\"evidence\":[],\"id\":\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"rule\":\"urn:stacks:clir:derived-functors#RFEverywhereDefined\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#derived_functors_form_delta_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:DerivedFormDeltaFunctor:a0ef0823d7ce3dfbf6ae694696e8274c0bcf0f1aa97d2c7f6441b07934ca990e\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#negative_derived_functors_vanish\"},\"evidence\":[],\"id\":\"urn:proof:apply:NegativeDerivedVanish:7fa0ea40bfa7150a31985ea95de55d4a32279727237bc17d6ac1afeb8319f335\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#NegativeDerivedVanish\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#r0_agrees_with_f\"},\"evidence\":[],\"id\":\"urn:proof:apply:R0AgreesIfLeftExact:1b60ec7124ec41ab9339140f079aff53eaf57168b5d3ea951596e08b137a0c4b\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#R0AgreesIfLeftExact\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-4\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-4\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-5\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_group_structure\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-5\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-6\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-6\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-7\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#gluing_is_unique\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-7\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafByGluing\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#object_of\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#has_injective_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:InjectiveResolutionsExist:92a8097774445945984ed4264ea6947dc499382a6154bd9e66dd68af3f4abaaf\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"rule\":\"urn:stacks:clir:derived-functors#InjectiveResolutionsExist\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-8\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#restrictions_surjective\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-8\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#flasque\"},\"evidence\":[],\"id\":\"urn:proof:apply:FlasqueByRestrictions:34f730ec0a1b8af2d0ae379c5a75880216a66b9739f6f1d5d50b2c80867dff8f\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-8\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#right_acyclic_for\"},\"evidence\":[],\"id\":\"urn:proof:apply:FlasqueIsAcyclic:eaa95f4954a00a93fffd4cad8ae80bb14882236dcfa2fc0ad1c73116fabe86e1\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:apply:FlasqueByRestrictions:34f730ec0a1b8af2d0ae379c5a75880216a66b9739f6f1d5d50b2c80867dff8f\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#higher_derived_vanish\"},\"evidence\":[],\"id\":\"urn:proof:apply:HigherDerivedVanishForAcyclic:6d47058a5fc5f3ad5543f1092b3393395165132d7b4775dfb2e8b828bea19ee8\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:FlasqueIsAcyclic:eaa95f4954a00a93fffd4cad8ae80bb14882236dcfa2fc0ad1c73116fabe86e1\",\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6\"],\"rule\":\"urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive\"},\"evidence\":[],\"id\":\"urn:proof:apply:CohomologyVanishesForAcyclic:a47cde96682458b0dac7ead3e82920921c7e4df46c52e53c52f25a310e17e8ce\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:HigherDerivedVanishForAcyclic:6d47058a5fc5f3ad5543f1092b3393395165132d7b4775dfb2e8b828bea19ee8\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"requirementStatus\":\"NEITHER\",\"status\":\"UNDETERMINED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"literal\":{\"args\":[{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive\"},\"truthStatus\":\"TRUE_ONLY\"},\"evidence\":[],\"id\":\"urn:proof:query:mcp\",\"kind\":\"query_evaluation\",\"premises\":[\"urn:proof:apply:CohomologyVanishesForAcyclic:a47cde96682458b0dac7ead3e82920921c7e4df46c52e53c52f25a310e17e8ce\"],\"sourceAnchors\":[]}],\"proofHash\":\"sha256:5d912b4bf938e63066eea6b355a3ab60755c59d0dce91da465e7f8033958330d\",\"roots\":[\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"urn:proof:query:mcp\"]},\"resultHash\":\"sha256:1166b2901cc0036291a26774c3e5724613084e2ab2033e1d286dca27e0aa25d3\",\"results\":[{\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:mcp\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatusSupports\":[],\"proof\":\"urn:proof:query:mcp\",\"query\":\"urn:query:mcp\",\"resultKind\":\"PROPOSITION\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_category/necessary:d7fa66f8c0f6f8c6\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_category/necessary:7a3e567cdcddb881\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:74ae6e0074519041\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[{\"description\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf: опора не установлена (§93.2)\",\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:74ae6e0074519041:input:0\",\"kind\":\"fact\",\"relatedNode\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf\"}],\"normativeStatus\":\"UNDETERMINED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"NEITHER\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_presheaf_on/necessary:3f36d6605863fecc\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.1\"}","evaluationSha256":"sha256:7b39be0f6824c88aa4f9fda1cf5aca51eaf6f428c57cf06e55d13007ad10c3a4","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-2","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-3","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-4","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-5","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-6","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-7","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-8","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#restrictions_surjective"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"}],"context":{"decisionTime":"2026-09-06T12:00:00+05:00","knowledgeTime":"2026-09-06T12:00:00+05:00","legalTime":"2026-09-06","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:8a58090dc9e51d0c78d83cb210fe43917cea71e0ee08f8f05532b76deb773e40","kind":"world_ref","nodeCount":383,"packages":[{"artifactHash":"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","semanticHash":"sha256:1ee2443e666f565715a3e3a5663914219d324c205aa6004f03c007d582687303"},{"artifactHash":"sha256:58778a4f746bcbfab9fdf432e1ba2b355fa02e4e0f2cefc1f41dec0f17f9cffc","namespace":"urn:stacks:clir:categories","package":"stacks-categories","semanticHash":"sha256:f21771fe43fa7adf28a188524efa50baebafb5892a4f79ebf8d717b9d3dcb1fc"},{"artifactHash":"sha256:99a2792dcc914bce6faddd3dca7875f19749b622e87c8e1f8a3007d8c6e6aaa3","namespace":"urn:stacks:clir:category-theory","package":"stacks-category-theory","semanticHash":"sha256:bd6536b4675c408f93bc08df24740211450e93a567c136fbad33194dfe200cc2"},{"artifactHash":"sha256:c522aaa206f22044b085107a030f77422be0f9478c88125c82a8f3b9ade621e3","namespace":"urn:stacks:clir:derived-functors","package":"stacks-derived-functors","semanticHash":"sha256:c699babe3fee3095884ef65f76edb67bf4c7c42c15f46f4e75ab10bc248922d7"}],"programHash":"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.1.0"},"requestCanonicalSha256":"sha256:d5086379c2535511f3863ae81a709ce2b9b3f02c8211c1bc00e8668cf00c88a8"},"id":"flasque-vanishing","kind":"truth","label":"Старшие когомологии вялого пучка нулевые","presentation":{"blockers":{"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","premises":[{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"missing":true,"predicate":"not_a_sheaf","predicateId":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","status":"NEITHER","text":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"}],"rule":"NotSheafByRefutingCovering","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering"}]},"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:d5086379c2535511f3863ae81a709ce2b9b3f02c8211c1bc00e8668cf00c88a8"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"id":"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"id":"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"id":"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"id":"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01DG"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"id":"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXKernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"id":"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":["urn:stacks:clir:categories#ST_0109"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Category"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"id":"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbAdditive","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"id":"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"id":"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCokernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"id":"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbKernels","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Category"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"id":"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":["urn:stacks:clir:categories#ST_0109"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Category"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"},{"kind":"var","var":"v2"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"id":"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0716"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Functor"}},{"id":"v1","type":{"name":"stacks.category_theory#Category"}},{"id":"v2","type":{"name":"stacks.category_theory#Category"}},{"id":"v3","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"id":"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0716"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"id":"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f"],"rule":"urn:stacks:clir:categories#LeftExactBySES","sourceAnchors":["urn:stacks:clir:categories#ST_010N"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Functor"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"id":"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd"],"rule":"urn:stacks:clir:categories#LeftExactIsAdditive","sourceAnchors":["urn:stacks:clir:categories#ST_010N"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Functor"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#rf_everywhere_defined"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#rf_everywhere_defined"},"id":"urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"rule":"urn:stacks:clir:derived-functors#RFEverywhereDefined","sourceAnchors":["urn:stacks:clir:derived-functors#ST_05TI","urn:stacks:clir:derived-functors#ST_05SV","urn:stacks:clir:derived-functors#ST_05T4"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:derived-functors#Category"}},{"id":"v2","type":{"name":"urn:stacks:clir:derived-functors#Category"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006K"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0070"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#restrictions_surjective"},"id":"urn:proof:assert:urn:mcp:case#fact-8","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"id":"urn:proof:apply:FlasqueByRestrictions:34f730ec0a1b8af2d0ae379c5a75880216a66b9739f6f1d5d50b2c80867dff8f","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-8"],"rule":"urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_09SW"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#right_acyclic_for"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#right_acyclic_for"},"id":"urn:proof:apply:FlasqueIsAcyclic:eaa95f4954a00a93fffd4cad8ae80bb14882236dcfa2fc0ad1c73116fabe86e1","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:apply:FlasqueByRestrictions:34f730ec0a1b8af2d0ae379c5a75880216a66b9739f6f1d5d50b2c80867dff8f","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_09SY","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"stacks.category_theory#Functor"}},{"id":"v2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#higher_derived_vanish"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#higher_derived_vanish"},"id":"urn:proof:apply:HigherDerivedVanishForAcyclic:6d47058a5fc5f3ad5543f1092b3393395165132d7b4775dfb2e8b828bea19ee8","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:FlasqueIsAcyclic:eaa95f4954a00a93fffd4cad8ae80bb14882236dcfa2fc0ad1c73116fabe86e1","urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","urn:proof:apply:RFEverywhereDefined:83675b793ceb474e07a4db088e9f5b9a2bda8668f9eb1c6104d247823b225ed6"],"rule":"urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic","sourceAnchors":["urn:stacks:clir:derived-functors#ST_015C"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:derived-functors#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:derived-functors#Obj"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive"},"head":{"args":[{"kind":"var","var":"v1"},{"kind":"var","var":"v2"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive"},"id":"urn:proof:apply:CohomologyVanishesForAcyclic:a47cde96682458b0dac7ead3e82920921c7e4df46c52e53c52f25a310e17e8ce","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:HigherDerivedVanishForAcyclic:6d47058a5fc5f3ad5543f1092b3393395165132d7b4775dfb2e8b828bea19ee8","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_09SY","urn:stacks:clir:sheaf-cohomology#ST_01DZ"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Functor"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"id":"v2","type":{"name":"stacks.category_theory#Obj"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:x","kind":"entity_ref"},{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:CohomologyVanishesForAcyclic:a47cde96682458b0dac7ead3e82920921c7e4df46c52e53c52f25a310e17e8ce"],"trust":"engine"}],"symbols":[{"id":"urn:stacks:clir:categories#LeftExactBySES","kind":"rule","labels":[{"language":"en","status":"official","text":"010N (2): \\(F\\) is left exact if 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":"010N (2): \\(F\\) точен слева, если для всякой короткой точной последовательности \\(0 → A → B → C → 0\\) последовательность \\(0 → F(A) → F(B) → F(C)\\) точна"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#LeftExactIsAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"010N (1): if \\(F\\) is left exact, then it is additive"},{"language":"ru","status":"unofficial","text":"010N (1): если \\(F\\) точен слева, то он аддитивен"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#abelian_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"name":"abelian_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#abelian_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#abelian_category/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"package":"urn:stacks:clir:categories"},{"id":"urn:stacks:clir:categories#abelian_category/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#additive_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0104: the category is additive"},{"language":"ru","status":"unofficial","text":"0104: категория аддитивна"}],"name":"additive_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#additive_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#additive_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is additive"},{"language":"ru","status":"unofficial","text":"функтор аддитивен"}],"name":"additive_functor","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#additive_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:categories#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#coimage_to_image_isomorphism","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the natural map \\(Coim(f) → Im(f)\\) is an isomorphism for all morphisms \\(f\\) of the 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#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#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#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"},{"id":"urn:stacks:clir:sheaf-cohomology#AbAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism for every morphism"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм для всякого морфизма"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","kind":"rule","labels":[{"language":"en","status":"official","text":"01DG: the category of abelian sheaves on a topological space \\(X\\) has enough injectives"},{"language":"ru","status":"unofficial","text":"01DG: в категории абелевых пучков на топологическом пространстве \\(X\\) достаточно инъективных"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (1): an abelian presheaf on \\(X\\) whose underlying presheaf of sets is a sheaf is an abelian sheaf"},{"language":"ru","status":"unofficial","text":"0070 (1): абелев предпучок на \\(X\\), чей предпучок множеств — пучок, есть абелев пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (2): an abelian sheaf on \\(X\\) is an object of \\(Ab(X)\\)"},{"language":"ru","status":"unofficial","text":"0070 (2): абелев пучок на \\(X\\) — объект категории \\(Ab(X)\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor","kind":"rule","labels":[{"language":"en","status":"official","text":"01DZ: the functors \\(H^i(X, −)\\) form a universal \\(δ\\)-functor \\(Ab(X) → Ab\\)"},{"language":"ru","status":"unofficial","text":"01DZ: функторы \\(H^i(X, −)\\) образуют универсальный \\(δ\\)-функтор \\(Ab(X) → Ab\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic","kind":"rule","labels":[{"language":"en","status":"official","text":"09SY with 015C (2): if \\(F\\) is right acyclic for the left exact \\(Γ(X, −)\\), then \\(H^i(X, F) = R^iΓ(X, −)(F) = 0\\) for \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"09SY с 015C (2): если \\(F\\) правый ацикличный для точного слева \\(Γ(X, −)\\), то \\(H^i(X, F) = R^iΓ(X, −)(F) = 0\\) при \\(i > 0\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions","kind":"rule","labels":[{"language":"en","status":"official","text":"09SW: a presheaf is flasque if all restriction maps \\(F(V) → F(U)\\), \\(U ⊂ V\\) open, are surjective"},{"language":"ru","status":"unofficial","text":"09SW: предпучок вялый, если все ограничения \\(F(V) → F(U)\\), \\(U ⊂ V\\) открытые, сюръективны"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","kind":"rule","labels":[{"language":"en","status":"official","text":"09SY with 01AD: a flasque abelian sheaf is right acyclic for \\(Γ(X, −)\\)"},{"language":"ru","status":"unofficial","text":"09SY с 01AD: вялый абелев пучок правый ацикличный для \\(Γ(X, −)\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a functor from \\(Ab(X)\\) to \\(Ab\\)"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — функтор из \\(Ab(X)\\) в \\(Ab\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a left exact functor"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — точный слева функтор"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","kind":"rule","labels":[{"language":"en","status":"official","text":"0FKS with 01AD: every abelian sheaf \\(F\\) on \\(X\\) has the Godement resolution \\(0 → F → f_*f^*F → …\\) by flasque sheaves"},{"language":"ru","status":"unofficial","text":"0FKS с 01AD: у всякого абелева пучка \\(F\\) на \\(X\\) есть резольвента Годемана \\(0 → F → f_*f^*F → …\\) вялыми пучками"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"006T: предпучок, опровергнутый открытым покрытием, — не пучок множеств на \\(X\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section"},{"language":"ru","status":"unofficial","text":"006T: предпучок множеств — пучок, если согласованные семейства сечений над любым открытым покрытием склеиваются в единственное сечение"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"each \\(F(U)\\) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups"},{"language":"ru","status":"unofficial","text":"каждое \\(F(U)\\) несёт структуру абелевой группы, и все отображения ограничения — гомоморфизмы абелевых групп"}],"name":"abelian_group_structure","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab\\), the category of abelian groups"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab\\) — категория абелевых групп"}],"name":"abelian_groups_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category/arg/ab","labels":[],"name":"ab","type":{"name":"stacks.category_theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"name":"abelian_presheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0070: \\(F\\) is an abelian sheaf on \\(X\\) — an abelian presheaf whose underlying presheaf of sets is a sheaf"},{"language":"ru","status":"unofficial","text":"0070: \\(F\\) — абелев пучок на \\(X\\): абелев предпучок, чей предпучок множеств — пучок"}],"name":"abelian_sheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"01DZ: the family \\(H^i(X, −)\\) forms a universal \\(δ\\)-functor from \\(Ab(X)\\) to \\(Ab\\)"},{"language":"ru","status":"unofficial","text":"01DZ: семейство \\(H^i(X, −)\\) образует универсальный \\(δ\\)-функтор из \\(Ab(X)\\) в \\(Ab\\)"}],"name":"cohomology_universal_delta_functor","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_universal_delta_functor/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(H^i(X, F) = 0\\) for all \\(i > 0\\)"},{"language":"ru","status":"unofficial","text":"\\(H^i(X, F) = 0\\) при всех \\(i > 0\\)"}],"name":"cohomology_vanishes_positive","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}},{"id":"urn:stacks:clir:sheaf-cohomology#cohomology_vanishes_positive/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, existence: for every open covering \\(U = ∪ U_i\\) and sections \\(s_i ∈ F(U_i)\\) with \\(s_i|U_i∩U_j = s_j|U_i∩U_j\\) there exists \\(s ∈ F(U)\\) with \\(s|U_i = s_i\\)"},{"language":"ru","status":"unofficial","text":"006T, существование: для всякого открытого покрытия \\(U = ∪ U_i\\) и сечений \\(s_i ∈ F(U_i)\\), согласованных на пересечениях, существует \\(s ∈ F(U)\\) с \\(s|U_i = s_i\\)"}],"name":"compatible_sections_glue","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#flasque","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"09SW: the sheaf is flasque (flabby)"},{"language":"ru","status":"unofficial","text":"09SW: пучок вялый (flasque)"}],"name":"flasque","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#flasque/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"},{"language":"ru","status":"unofficial","text":"функтор есть \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"}],"name":"global_sections_functor","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Functor"}},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, uniqueness: a section \\(s ∈ F(U)\\) is determined by its restrictions \\(s|U_i\\) to an open covering"},{"language":"ru","status":"unofficial","text":"006T, единственность: сечение \\(s ∈ F(U)\\) определяется своими ограничениями \\(s|U_i\\) на открытое покрытие"}],"name":"gluing_is_unique","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)"},{"language":"ru","status":"unofficial","text":"0FKS: у пучка есть функториальная резольвента вялыми пучками (резольвента Годемана)"}],"name":"has_flasque_resolution","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is not a sheaf on \\(X\\): the sheaf condition 006T fails on some open covering"},{"language":"ru","status":"unofficial","text":"\\(F\\) — не пучок на \\(X\\): условие 006T провалено на некотором открытом покрытии"}],"name":"not_a_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a presheaf of sets on \\(X\\): a rule assigning a set \\(F(U)\\) to each open \\(U\\) and restriction maps \\(ρ^U_V\\) to inclusions \\(V ⊂ U\\) with \\(ρ^U_U = id\\) and \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — предпучок множеств на \\(X\\): правило, сопоставляющее каждому открытому \\(U\\) множество \\(F(U)\\), а вложениям \\(V ⊂ U\\) — отображения ограничения \\(ρ^U_V\\) с \\(ρ^U_U = id\\) и \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"}],"name":"presheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#restrictions_surjective","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"09SW: for every \\(U ⊂ V\\) open in \\(X\\) the restriction map \\(F(V) → F(U)\\) is surjective"},{"language":"ru","status":"unofficial","text":"09SW: для всяких открытых \\(U ⊂ V\\) в \\(X\\) отображение ограничения \\(F(V) → F(U)\\) сюръективно"}],"name":"restrictions_surjective","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#restrictions_surjective/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — пучок множеств на \\(X\\)"}],"name":"sheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab(X)\\), the category of sheaves of abelian groups on \\(X\\)"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab(X)\\) — категория пучков абелевых групп на \\(X\\)"}],"name":"sheaves_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/c","labels":[],"name":"c","type":{"name":"stacks.category_theory#Category"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_01DZ","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_09SW","urn:stacks:clir:sheaf-cohomology#ST_09SY","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_0109","urn:stacks:clir:categories#ST_010N"],"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"},{"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_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"}],"caseHash":"sha256:eb097105acd6264efc4b3a976119c21291db893e9fd17e38d38569e961f8c9fa","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:037863663edcbb19699db703e2b64c892fac09e4362d28c4147d70de0b0fb692","resultHash":"sha256:1166b2901cc0036291a26774c3e5724613084e2ab2033e1d286dca27e0aa25d3","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:sh:x","urn:case:stacks:sh:f"],"facts":[{"args":["urn:case:stacks:sh:abx","urn:case:stacks:sh:x"],"predicate":"sheaves_category"},{"args":["urn:case:stacks:sh:ab"],"predicate":"abelian_groups_category"},{"args":["urn:case:stacks:sh:gamma","urn:case:stacks:sh:x"],"predicate":"global_sections_functor"},{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"predicate":"presheaf_of_sets_on"},{"args":["urn:case:stacks:sh:f"],"predicate":"abelian_group_structure"},{"args":["urn:case:stacks:sh:f"],"predicate":"compatible_sections_glue"},{"args":["urn:case:stacks:sh:f"],"predicate":"gluing_is_unique"},{"args":["urn:case:stacks:sh:f"],"predicate":"restrictions_surjective"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"cohomology_vanishes_positive","proof":true},"sources":[{"contentHash":"sha256:fa7c3dec36920848465ed2902237f8efade0877767eebcee37676bb1bb8717d9","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006K","kind":"fragment","locator":"tag/006K","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:b8388b54fb29de485ebd7c91f38c607f6005d41b4d427265a75b163118fd70d6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-presheaves}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf of abelian groups on $X$} or an\n{\\it abelian presheaf over $X$}\nis a presheaf of sets $\\mathcal{F}$ such that for each open\n$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with\nthe structure of an abelian group, and such that all restriction\nmaps $\\rho^U_V$ are homomorphisms of abelian groups, see\nLemma \\ref{lemma-abelian-presheaves} above.\n\\item A {\\it morphism of abelian presheaves over $X$}\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves\nof sets which induces\na homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nfor every open $U \\subset X$.\n\\item The category of presheaves of abelian groups on $X$ is denoted\n$\\textit{PAb}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:8504f30501fb40b30a60685a51aad76a416258206ec9a839891f7bf83cd38323","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006T","kind":"fragment","locator":"tag/006T","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:04f51a87121abeee45cd4e99f1379db299b359b4ac801017c7770a7f05d2c370","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_0070","kind":"fragment","locator":"tag/0070","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3260f7d5e9cbfe266160539b8375cd23ac2429762aaa12eb860ad707dc28a6cd","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item An {\\it abelian sheaf on $X$} or\n{\\it sheaf of abelian groups on $X$}\nis an abelian presheaf on $X$ such that the underlying presheaf of\nsets is a sheaf.\n\\item The category of sheaves of abelian groups\nis denoted $\\textit{Ab}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:d0262bf7656850a1933eef167a360437aa2bb6436e57461a3d23de09e8cc8ef3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01AD","kind":"fragment","locator":"tag/01AD","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf","language":"en","status":"official","text":"\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we work out basic notions of sheaves of modules.\nThis in particular includes the case of abelian sheaves, since\nthese may be viewed as sheaves of $\\underline{\\mathbf{Z}}$-modules.\nBasic references are \\cite{FAC}, \\cite{EGA} and \\cite{SGA4}.\n\n\\medskip\\noindent\nWe work out what happens for sheaves of modules on ringed topoi\nin another chapter (see\nModules on Sites, Section \\ref{sites-modules-section-introduction}),\nalthough there we will mostly just duplicate the discussion\nfrom this chapter."}]},{"contentHash":"sha256:1bf6bd6db42698470b675022b39392436590d133c6e4e0e6f342e63ce50e5290","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01AG","kind":"fragment","locator":"tag/01AG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:d40ca7777a52e10948b9c7e8da68432632fc0b8f4bb33c427370acaf48a4b19f","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover\na complex\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nis exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the\ncomplex\n$$\n\\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x\n$$\nis exact at $\\mathcal{G}_x$.\n\\end{lemma}"}]},{"contentHash":"sha256:48de83780b89577a36cf370ecf47370fa86fe2d17276908d3fb411375d5e1ec7","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_INJECTIVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01DG","kind":"fragment","locator":"tag/01DG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e4b4f09acb0a39efab590f7a0a1974dde000a27f6205d5284eb9b41b91b7f6d6","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian-sheaves-space}\nLet $X$ be a topological space.\nThe category of abelian sheaves on $X$ has enough injectives.\nIn fact it has functorial injective embeddings.\n\\end{lemma}"}]},{"contentHash":"sha256:b1dd6e78bfe8178dd35b60b16af0e740bb77f76f64096d01aaa44a811d4491b3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01DZ","kind":"fragment","locator":"tag/01DZ","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:992dfb482cc74c054eb720fd53a56dd386827983f1cb4214bf3e7230a56ea9d3","language":"en","status":"official","text":"\\section{Cohomology of sheaves}\n\\label{section-cohomology-sheaves}\n\n\\noindent\nLet $X$ be a topological space. Let $\\mathcal{F}$ be an abelian sheaf.\nWe know that the category of abelian sheaves on $X$ has enough injectives, see\nInjectives, Lemma \\ref{injectives-lemma-abelian-sheaves-space}.\nHence we can choose an injective resolution\n$\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$. As is customary we define\n\\begin{equation}\n\\label{equation-cohomology}\nH^i(X, \\mathcal{F}) = H^i(\\Gamma(X, \\mathcal{I}^\\bullet))\n\\end{equation}\nto be the {\\it $i$th cohomology group of the abelian sheaf $\\mathcal{F}$}.\nThe family of functors $H^i(X, -)$ forms a universal $\\delta$-functor\nfrom $\\textit{Ab}(X) \\to \\textit{Ab}$.\n\n\\medskip\\noindent\nLet $f : X \\to Y$ be a continuous map of topological spaces. With\n$\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$ as above\nwe define\n\\begin{equation}\n\\label{equation-higher-direct-image}\nR^if_*\\mathcal{F} = H^i(f_*\\mathcal{I}^\\bullet)\n\\end{equation}\nto be the {\\it $i$th higher direct image of $\\mathcal{F}$}.\nThe family of functors $R^if_*$ forms a universal $\\delta$-functor\nfrom $\\textit{Ab}(X) \\to \\textit{Ab}(Y)$.\n\n\\medskip\\noindent\nLet $(X, \\mathcal{O}_X)$ be a ringed space. Let $\\mathcal{F}$ be an\n$\\mathcal{O}_X$-module. We know that the category of $\\mathcal{O}_X$-modules\non $X$ has enough injectives, see\nInjectives, Lemma \\ref{injectives-lemma-sheaves-modules-space}.\nHence we can choose an injective resolution\n$\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$. As is customary we define\n\\begin{equation}\n\\label{equation-cohomology-modules}\nH^i(X, \\mathcal{F}) = H^i(\\Gamma(X, \\mathcal{I}^\\bullet))\n\\end{equation}\nto be the {\\it $i$th cohomology group of $\\mathcal{F}$}.\nThe family of functors $H^i(X, -)$ forms a universal $\\delta$-functor\nfrom $\\textit{Mod}(\\mathcal{O}_X) \\to \\text{Mod}_{\\mathcal{O}_X(X)}$.\n\n\\medskip\\noindent\nLet $f : (X, \\mathcal{O}_X) \\to (Y, \\mathcal{O}_Y)$ be a morphism of ringed\nspaces. With $\\mathcal{F}[0] \\to \\mathcal{I}^\\bullet$ as above\nwe define\n\\begin{equation}\n\\label{equation-higher-direct-image-modules}\nR^if_*\\mathcal{F} = H^i(f_*\\mathcal{I}^\\bullet)\n\\end{equation}\nto be the {\\it $i$th higher direct image of $\\mathcal{F}$}.\nThe family of functors $R^if_*$ forms a universal $\\delta$-functor\nfrom $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$."}]},{"contentHash":"sha256:0eee6ff9b997a29e65a9204cd10ad0d6c3337360eac630526713e3e18f1c9b03","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_0716","kind":"fragment","locator":"tag/0716","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:f61344b89b5447a2349b548070137ca4e6f491c90033bf637527fbd9f371c876","language":"en","status":"official","text":"\\section{Derived functors}\n\\label{section-derived-functors}\n\n\\noindent\nWe briefly explain how to get right derived functors using resolution\nfunctors. For the unbounded derived functors, please see\nSection \\ref{section-unbounded}.\n\n\\medskip\\noindent\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is abelian, see\nModules, Lemma \\ref{modules-lemma-abelian}.\nIn this chapter we will write\n$$\nK(\\mathcal{O}_X) = K(\\textit{Mod}(\\mathcal{O}_X))\n\\quad\n\\text{and}\n\\quad\nD(\\mathcal{O}_X) = D(\\textit{Mod}(\\mathcal{O}_X)).\n$$\nand similarly for the bounded versions for the triangulated categories\nintroduced in\nDerived Categories, Definition \\ref{derived-definition-complexes-notation} and\nDefinition \\ref{derived-definition-unbounded-derived-category}.\nBy\nDerived Categories, Remark \\ref{derived-remark-big-abelian-category}\nthere exists a resolution functor\n$$\nj = j_X :\nK^{+}(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK^{+}(\\mathcal{I})\n$$\nwhere $\\mathcal{I}$ is the strictly full additive subcategory of\n$\\textit{Mod}(\\mathcal{O}_X)$ consisting of injective sheaves.\nFor any left exact functor\n$F : \\textit{Mod}(\\mathcal{O}_X) \\to \\mathcal{B}$\ninto any abelian category $\\mathcal{B}$ we will denote $RF$ the\nright derived functor described in\nDerived Categories, Section \\ref{derived-section-right-derived-functor}\nand constructed using the resolution functor $j_X$ just described:\n\\begin{equation}\n\\label{equation-RF}\nRF = F \\circ j_X' : D^{+}(X) \\longrightarrow D^{+}(\\mathcal{B})\n\\end{equation}\nsee\nDerived Categories, Lemma \\ref{derived-lemma-right-derived-functor}\nfor notation. Note that we may think of $RF$ as defined on\n$\\textit{Mod}(\\mathcal{O}_X)$,\n$\\text{Comp}^{+}(\\textit{Mod}(\\mathcal{O}_X))$,\n$K^{+}(X)$, or $D^{+}(X)$\ndepending on the situation. According to\nDerived Categories, Definition \\ref{derived-definition-higher-derived-functors}\nwe obtain the $i$th right derived functor\n\\begin{equation}\n\\label{equation-RFi}\nR^iF = H^i \\circ RF : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\mathcal{B}\n\\end{equation}\nso that $R^0F = F$ and $\\{R^iF, \\delta\\}_{i \\geq 0}$ is universal\n$\\delta$-functor, see\nDerived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}.\n\n\\medskip\\noindent\nHere are two special cases of this construction.\nGiven a ring $R$ we write $K(R) = K(\\text{Mod}_R)$ and\n$D(R) = D(\\text{Mod}_R)$ and similarly for bounded versions.\nFor any open $U \\subset X$ we have a left exact functor\n$\n\\Gamma(U, -) :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\text{Mod}_{\\mathcal{O}_X(U)}\n$\nwhich gives rise to\n\\begin{equation}\n\\label{equation-total-derived-cohomology}\nR\\Gamma(U, -) :\nD^{+}(X)\n\\longrightarrow\nD^{+}(\\mathcal{O}_X(U))\n\\end{equation}\nby the discussion above. We set $H^i(U, -) = R^i\\Gamma(U, -)$.\nIf $U = X$ we recover (\\ref{equation-cohomology-modules}).\nIf $f : X \\to Y$ is a morphism of ringed spaces, then we have\nthe left exact functor\n$\nf_* :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_Y)\n$\nwhich gives rise to the {\\it derived pushforward}\n\\begin{equation}\n\\label{equation-total-derived-direct-image}\nRf_* :\nD^{+}(X)\n\\longrightarrow\nD^{+}(Y)\n\\end{equation}\nThe $i$th cohomology sheaf of $Rf_*\\mathcal{F}^\\bullet$ is denoted\n$R^if_*\\mathcal{F}^\\bullet$ and called the $i$th {\\it higher direct image}\nin accordance with (\\ref{equation-higher-direct-image-modules}).\nThe two displayed functors above are exact functors\nof derived categories.\n\n\\medskip\\noindent\n{\\bf Abuse of notation:} When the functor $Rf_*$, or any other\nderived functor, is applied to a sheaf $\\mathcal{F}$ on $X$ or a complex\nof sheaves it is understood that $\\mathcal{F}$ has been replaced by a\nsuitable resolution of $\\mathcal{F}$. To facilitate this kind of\noperation we will say, given an object\n$\\mathcal{F}^\\bullet \\in D(\\mathcal{O}_X)$,\nthat a bounded below complex $\\mathcal{I}^\\bullet$ of injectives of\n$\\textit{Mod}(\\mathcal{O}_X)$\n{\\it represents $\\mathcal{F}^\\bullet$ in the derived category}\nif there exists a quasi-isomorphism\n$\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$. In the same vein the phrase\n``let $\\alpha : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be\na morphism of $D(\\mathcal{O}_X)$''\ndoes not mean that $\\alpha$ is represented by a\nmorphism of complexes. If we have an actual morphism of complexes we will\nsay so."}]},{"contentHash":"sha256:fe2dbc1108dafb7562aec854fa649c8fc23820c9279877f4cb4d498b8601d0e7","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_09SW","kind":"fragment","locator":"tag/09SW","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:730a7c4f799967f2799cc62798491a402948504907086636e81743211b35b8b0","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-flasque}\nLet $X$ be a topological space. We say a presheaf of sets\n$\\mathcal{F}$ is {\\it flasque} or {\\it flabby} if for every\n$U \\subset V$ open in $X$ the restriction map\n$\\mathcal{F}(V) \\to \\mathcal{F}(U)$ is surjective.\n\\end{definition}"}]},{"contentHash":"sha256:7662615771c020c4afb51a8478766a67d4a3bdf502078614c0096d548d27dc05","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_09SY","kind":"fragment","locator":"tag/09SY","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:707ef368c24b80d3d736e07759dff795af4faae92c0ece2f17999e27898ff0d2","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-flasque-acyclic}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. Any flasque $\\mathcal{O}_X$-module\nis acyclic for $R\\Gamma(X, -)$ as well as $R\\Gamma(U, -)$ for any\nopen $U$ of $X$.\n\\end{lemma}"}]},{"contentHash":"sha256:787503a0ec46db68e4e11e9bf5fa2fabae3304c806bc7e2f5be69ab1af6cc11f","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0FKS","kind":"fragment","locator":"tag/0FKS","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:8b044713268f8366ca6cc0212d88985ff0517eb2417f464d218f454e93c283fe","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-godement-resolution}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution\n$$\n0 \\to\n\\mathcal{F} \\to\nf_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots\n$$\nfunctorial in $\\mathcal{F}$ such that each term\n$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque\n$\\mathcal{O}_X$-module and such that for all $x \\in X$ the\nmap\n$$\n\\mathcal{F}_x[0] \\to \\Big(\n(f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots\n\\Big)\n$$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.\n\\end{lemma}"}]},{"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":"sha256:9698ad04b03a0ca888a87e8f88e9925a856764046ae30cfb569730f33de5c92d","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":"sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e","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}"}]}],"text":"cohomology_vanishes_positive: **TRUE_ONLY** — установлено\nВыведено правом: cohomology_universal_delta_functor(urn:case:stacks:sh:x); abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx); has_flasque_resolution(urn:case:stacks:sh:f); has_injective_resolution(urn:case:stacks:sh:f); flasque(urn:case:stacks:sh:f); right_acyclic_for(urn:case:stacks:sh:f, urn:case:stacks:sh:gamma); higher_derived_vanish(urn:case:stacks:sh:gamma, urn:case:stacks:sh:f); cohomology_vanishes_positive(urn:case:stacks:sh:x, urn:case:stacks:sh:f)\n…и ещё 20 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, HigherDerivedVanishForAcyclic, InjectiveResolutionsExist, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined, AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXEnoughInjectives, AbXKernels, AbelianSheafByDefinition, AbelianSheafIsObjectOfAbX, CohomologyUniversalDeltaFunctor, CohomologyVanishesForAcyclic, FlasqueByRestrictions, FlasqueIsAcyclic, GlobalSectionsBetween, GlobalSectionsLeftExact, GodementResolutionExists, SheafByGluing, abelian_presheaf_on/sufficient\nОтвет поражаем правилом «006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)» — оно отменило бы вывод, будь установлено: not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:037863663edc…)\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; Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина\nproof-граф: 44 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx)","has_flasque_resolution(urn:case:stacks:sh:f)","injective_object(urn:case:stacks:sh:f)","flasque(urn:case:stacks:sh:f)","right_acyclic_for(urn:case:stacks:sh:f, urn:case:stacks:sh:gamma)"],"derivedOmitted":15,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"evidence":[],"id":"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"definition":{"concept":"urn:stacks:clir:categories#abelian_category","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"evidence":[],"id":"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372","kind":"rule_application","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-2"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"definition":{"concept":"urn:stacks:clir:categories#abelian_category","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#abelian_category"},"evidence":[],"id":"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0","kind":"rule_application","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61"],"rule":"urn:stacks:clir:categories#abelian_category/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-3"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#left_exact_functor"},"evidence":[],"id":"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd","kind":"rule_application","premises":["urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f"],"rule":"urn:stacks:clir:categories#LeftExactBySES","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_functor"},"evidence":[],"id":"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a","kind":"rule_application","premises":["urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd"],"rule":"urn:stacks:clir:categories#LeftExactIsAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-4"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-5"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"definition":{"concept":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"evidence":[],"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-6"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-7"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"evidence":[],"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"evidence":[],"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#object_of"},"evidence":[],"id":"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"evidence":[],"id":"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b"],"rule":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-8"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#lifting_property"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-8","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#injective_object"},"evidence":[],"id":"urn:proof:apply:InjectiveByLifting:18dff9b5f33678b9bf2a8dda453cbaba3d1da8c15ad78436274455cea88d1239","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-8"],"rule":"urn:stacks:clir:categories#InjectiveByLifting","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"evidence":[],"id":"urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","urn:proof:apply:InjectiveByLifting:18dff9b5f33678b9bf2a8dda453cbaba3d1da8c15ad78436274455cea88d1239","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:derived-functors#right_acyclic_for"},"evidence":[],"id":"urn:proof:apply:FlasqueIsAcyclic:2d5aee08b3739b4a1003ca2eef56bb76c05b3b68e61bbe848d48efd2aaf75b12","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","kind":"constraint_check","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:categories#abelian_category/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585","kind":"constraint_check","premises":["urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:categories#abelian_category/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749","kind":"constraint_check","premises":["urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","kind":"constraint_check","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339"],"sourceAnchors":[]}],"proofHash":"sha256:c75db44872b2da89eab5f5fefdb6f644ac7a80380fcb165d3a0020649d1af781","roots":["urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585","urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749","urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","urn:proof:query:mcp"]},"resultHash":"sha256:d587a046da0ef9a29dc3ed4d1f6bc4885af175d52f25c2a1767aa7fa1b404792","schemaVersion":"law.core.evaluation/0.1"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_09SX","urn:stacks:clir:sheaf-cohomology#ST_09SY","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_0109","urn:stacks:clir:categories#ST_010N","urn:stacks:clir:categories#ST_0135"],"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"}],"caseHash":"sha256:bbbb3a110161e09d0f7a6e2793f2b776e0dfbb7278bbd7c430796e7468451fde","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:3a0bacb658a20b29b5d6e7bd7e3fb02f592fc0d840ab878384bb7cfe11ae5aba","resultHash":"sha256:d587a046da0ef9a29dc3ed4d1f6bc4885af175d52f25c2a1767aa7fa1b404792","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"rulesApplied":["urn:stacks:clir:categories#InjectiveByLifting","urn:stacks:clir:categories#LeftExactBySES","urn:stacks:clir:categories#LeftExactIsAdditive","urn:stacks:clir:categories#abelian_category/sufficient","urn:stacks:clir:sheaf-cohomology#AbAdditive","urn:stacks:clir:sheaf-cohomology#AbCoimageImage","urn:stacks:clir:sheaf-cohomology#AbCokernels","urn:stacks:clir:sheaf-cohomology#AbKernels","urn:stacks:clir:sheaf-cohomology#AbXAdditive","urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","urn:stacks:clir:sheaf-cohomology#AbXCokernels","urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","urn:stacks:clir:sheaf-cohomology#AbXKernels","urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque","urn:stacks:clir:sheaf-cohomology#SheafByGluing","urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient"],"signature":{"constants":{},"parameters":[{"labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Obj"}}],"predicate":"urn:stacks:clir:sheaf-cohomology#flasque","schemaVersion":"law.answers.signature/0.1","types":{"urn:stacks:clir:category-theory#Obj":{"kind":"unknown"}},"vocab":{}},"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","missing":["not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"],"premises":[{"premise":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByRefutingCovering"}],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:bbbb3a110161e09d0f7a6e2793f2b776e0dfbb7278bbd7c430796e7468451fde\",\"decisionTime\":\"2026-09-06T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-09-06T12:00:00+05:00\",\"legalTime\":\"2026-09-06\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:3a0bacb658a20b29b5d6e7bd7e3fb02f592fc0d840ab878384bb7cfe11ae5aba\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:82363c56458b204b2f0641257f88c0a2074408baef054af1e03e52440913b425\",\"semantics\":\"law.core/0.1.0\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaves_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#enough_injectives\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:categories#abelian_category\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#abelian_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\"],\"rule\":\"urn:stacks:clir:categories#abelian_category/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-2\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_groups_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-2\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:categories#abelian_category\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#abelian_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\"],\"rule\":\"urn:stacks:clir:categories#abelian_category/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-3\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#global_sections_functor\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-3\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#functor_between\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"},\"v3\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#preserves_left_exactness\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#left_exact_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\"],\"rule\":\"urn:stacks:clir:categories#LeftExactBySES\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_functor\"},\"evidence\":[],\"id\":\"urn:proof:apply:LeftExactIsAdditive:9450c71e95013c7152aea71217286b87d8afddfc372fe9f2dfcf8a4dffcab71a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:LeftExactBySES:2683e347b43daa7f29bdfc41d1e8b16443844795e2941f43ad82193ebd84d5cd\"],\"rule\":\"urn:stacks:clir:categories#LeftExactIsAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-4\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-4\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-5\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_group_structure\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-5\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-6\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-6\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-7\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#gluing_is_unique\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-7\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafByGluing\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#object_of\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-8\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#lifting_property\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-8\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#injective_object\"},\"evidence\":[],\"id\":\"urn:proof:apply:InjectiveByLifting:18dff9b5f33678b9bf2a8dda453cbaba3d1da8c15ad78436274455cea88d1239\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-8\"],\"rule\":\"urn:stacks:clir:categories#InjectiveByLifting\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#flasque\"},\"evidence\":[],\"id\":\"urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"urn:proof:apply:InjectiveByLifting:18dff9b5f33678b9bf2a8dda453cbaba3d1da8c15ad78436274455cea88d1239\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:derived-functors#right_acyclic_for\"},\"evidence\":[],\"id\":\"urn:proof:apply:FlasqueIsAcyclic:2d5aee08b3739b4a1003ca2eef56bb76c05b3b68e61bbe848d48efd2aaf75b12\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"requirementStatus\":\"NEITHER\",\"status\":\"UNDETERMINED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"urn:proof:apply:abelian_category/sufficient:cdc6bbb810a576d39fe7b0559bf317cbc71a1f9b7f7098d0a3dc022ee5733372\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"urn:proof:apply:abelian_category/sufficient:0438497c5c8cb1be1d93cc85eb712053b070b6670034893baec476363029e7f0\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"literal\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#flasque\"},\"truthStatus\":\"TRUE_ONLY\"},\"evidence\":[],\"id\":\"urn:proof:query:mcp\",\"kind\":\"query_evaluation\",\"premises\":[\"urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339\"],\"sourceAnchors\":[]}],\"proofHash\":\"sha256:c75db44872b2da89eab5f5fefdb6f644ac7a80380fcb165d3a0020649d1af781\",\"roots\":[\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"urn:proof:query:mcp\"]},\"resultHash\":\"sha256:d587a046da0ef9a29dc3ed4d1f6bc4885af175d52f25c2a1767aa7fa1b404792\",\"results\":[{\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:mcp\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatusSupports\":[],\"proof\":\"urn:proof:query:mcp\",\"query\":\"urn:query:mcp\",\"resultKind\":\"PROPOSITION\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#flasque\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_category/necessary:5a89bd151322e743\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_category/necessary:7bd94300abe4ec3a5e827c7aa7767e3e8f1e49beb10288d180a510fb515b6749\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_category/necessary:2b12568dd38f6e6a\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_category/necessary:0cd533fdf178fd64a4eb5848e724336d1d82e7b26114d75558e52a34eda0d585\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:categories#abelian_category/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:02a7d35f107e2da6\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[{\"description\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf: опора не установлена (§93.2)\",\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:02a7d35f107e2da6:input:0\",\"kind\":\"fact\",\"relatedNode\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf\"}],\"normativeStatus\":\"UNDETERMINED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"NEITHER\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_presheaf_on/necessary:8d12af88ad116c35\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.1\"}","evaluationSha256":"sha256:de1183bf01a85c57704d5ef59e996ed95975546c138645cf0aa651373ac6100e","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-2","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-3","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-4","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-5","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-6","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-7","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-8","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#lifting_property"},"origin":"case_input","package":"urn:stacks:clir:categories"}],"context":{"decisionTime":"2026-09-06T12:00:00+05:00","knowledgeTime":"2026-09-06T12:00:00+05:00","legalTime":"2026-09-06","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:ee17e531b2e8f38614979bd15ae9a7f4e4e3fd6119716cba3633e59ab84e026e","kind":"world_ref","nodeCount":226,"packages":[{"artifactHash":"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","semanticHash":"sha256:1ee2443e666f565715a3e3a5663914219d324c205aa6004f03c007d582687303"},{"artifactHash":"sha256:58778a4f746bcbfab9fdf432e1ba2b355fa02e4e0f2cefc1f41dec0f17f9cffc","namespace":"urn:stacks:clir:categories","package":"stacks-categories","semanticHash":"sha256:f21771fe43fa7adf28a188524efa50baebafb5892a4f79ebf8d717b9d3dcb1fc"}],"programHash":"sha256:3a0bacb658a20b29b5d6e7bd7e3fb02f592fc0d840ab878384bb7cfe11ae5aba"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.1.0"},"requestCanonicalSha256":"sha256:a4aa6c91c8d1224a67dc6615295bc05b1bae501b98297f40a84663085f91ee0e"},"id":"injective-flasque","kind":"truth","label":"Инъективный пучок вял","presentation":{"blockers":{"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","premises":[{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"missing":true,"predicate":"not_a_sheaf","predicateId":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","status":"NEITHER","text":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"}],"rule":"NotSheafByRefutingCovering","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering"}]},"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:a4aa6c91c8d1224a67dc6615295bc05b1bae501b98297f40a84663085f91ee0e"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006K"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0070"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#object_of"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#object_of"},"id":"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0070"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"stacks.category_theory#Category"}},{"id":"v2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#lifting_property"},"id":"urn:proof:assert:urn:mcp:case#fact-8","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#injective_object"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#injective_object"},"id":"urn:proof:apply:InjectiveByLifting:18dff9b5f33678b9bf2a8dda453cbaba3d1da8c15ad78436274455cea88d1239","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-8"],"rule":"urn:stacks:clir:categories#InjectiveByLifting","sourceAnchors":["urn:stacks:clir:categories#ST_0135"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"urn:stacks:clir:categories#Obj"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"id":"urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","urn:proof:apply:InjectiveByLifting:18dff9b5f33678b9bf2a8dda453cbaba3d1da8c15ad78436274455cea88d1239","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_09SX","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"stacks.category_theory#Category"}},{"id":"v2","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#flasque"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:InjectiveSheafIsFlasque:c29d6f3514f5c7411120c8567f4e6c9943f2607e94830110315df4f87ddcf339"],"trust":"engine"}],"symbols":[{"id":"urn:stacks:clir:categories#InjectiveByLifting","kind":"rule","labels":[{"language":"en","status":"official","text":"0135: an object \\(J\\) is injective if every morphism \\(A → J\\) extends along every injection \\(A ↪ B\\)"},{"language":"ru","status":"unofficial","text":"0135: объект \\(J\\) инъективен, если всякий морфизм \\(A → J\\) продолжается вдоль всякой инъекции \\(A ↪ B\\)"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#LeftExactBySES","kind":"rule","labels":[{"language":"en","status":"official","text":"010N (2): \\(F\\) is left exact if 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":"010N (2): \\(F\\) точен слева, если для всякой короткой точной последовательности \\(0 → A → B → C → 0\\) последовательность \\(0 → F(A) → F(B) → F(C)\\) точна"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#LeftExactIsAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"010N (1): if \\(F\\) is left exact, then it is additive"},{"language":"ru","status":"unofficial","text":"010N (1): если \\(F\\) точен слева, то он аддитивен"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#abelian_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"name":"abelian_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#abelian_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#abelian_category/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"package":"urn:stacks:clir:categories"},{"id":"urn:stacks:clir:categories#abelian_category/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"0109: a category is abelian if it is additive, all kernels and cokernels exist, and \\(Coim(f) → Im(f)\\) is an isomorphism for all \\(f\\)"},{"language":"ru","status":"unofficial","text":"0109: категория абелева, если она аддитивна, в ней существуют все ядра и коядра и \\(Coim(f) → Im(f)\\) — изоморфизм для всех \\(f\\)"}],"package":"urn:stacks:clir:categories","strength":"strict"},{"id":"urn:stacks:clir:categories#additive_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0104: the category is additive"},{"language":"ru","status":"unofficial","text":"0104: категория аддитивна"}],"name":"additive_category","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#additive_category/arg/c","labels":[],"name":"c","type":{"name":"urn:stacks:clir:categories#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#additive_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is additive"},{"language":"ru","status":"unofficial","text":"функтор аддитивен"}],"name":"additive_functor","package":"urn:stacks:clir:categories","parameters":[{"id":"urn:stacks:clir:categories#additive_functor/arg/f","labels":[],"name":"f","type":{"name":"urn:stacks:clir:categories#Functor"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:categories#coimage_to_image_isomorphism","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the natural map \\(Coim(f) → Im(f)\\) is an isomorphism for all morphisms \\(f\\) of the 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#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#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:sheaf-cohomology#AbAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism for every morphism"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм для всякого морфизма"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","kind":"rule","labels":[{"language":"en","status":"official","text":"01DG: the category of abelian sheaves on a topological space \\(X\\) has enough injectives"},{"language":"ru","status":"unofficial","text":"01DG: в категории абелевых пучков на топологическом пространстве \\(X\\) достаточно инъективных"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (1): an abelian presheaf on \\(X\\) whose underlying presheaf of sets is a sheaf is an abelian sheaf"},{"language":"ru","status":"unofficial","text":"0070 (1): абелев предпучок на \\(X\\), чей предпучок множеств — пучок, есть абелев пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (2): an abelian sheaf on \\(X\\) is an object of \\(Ab(X)\\)"},{"language":"ru","status":"unofficial","text":"0070 (2): абелев пучок на \\(X\\) — объект категории \\(Ab(X)\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic","kind":"rule","labels":[{"language":"en","status":"official","text":"09SY with 01AD: a flasque abelian sheaf is right acyclic for \\(Γ(X, −)\\)"},{"language":"ru","status":"unofficial","text":"09SY с 01AD: вялый абелев пучок правый ацикличный для \\(Γ(X, −)\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a functor from \\(Ab(X)\\) to \\(Ab\\)"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — функтор из \\(Ab(X)\\) в \\(Ab\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a left exact functor"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — точный слева функтор"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","kind":"rule","labels":[{"language":"en","status":"official","text":"0FKS with 01AD: every abelian sheaf \\(F\\) on \\(X\\) has the Godement resolution \\(0 → F → f_*f^*F → …\\) by flasque sheaves"},{"language":"ru","status":"unofficial","text":"0FKS с 01AD: у всякого абелева пучка \\(F\\) на \\(X\\) есть резольвента Годемана \\(0 → F → f_*f^*F → …\\) вялыми пучками"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque","kind":"rule","labels":[{"language":"en","status":"official","text":"09SX with 01AD: an injective object of \\(Ab(X) = Mod(Z_X)\\) is flasque"},{"language":"ru","status":"unofficial","text":"09SX с 01AD: инъективный объект \\(Ab(X) = Mod(Z_X)\\) вял"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"006T: предпучок, опровергнутый открытым покрытием, — не пучок множеств на \\(X\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section"},{"language":"ru","status":"unofficial","text":"006T: предпучок множеств — пучок, если согласованные семейства сечений над любым открытым покрытием склеиваются в единственное сечение"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"each \\(F(U)\\) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups"},{"language":"ru","status":"unofficial","text":"каждое \\(F(U)\\) несёт структуру абелевой группы, и все отображения ограничения — гомоморфизмы абелевых групп"}],"name":"abelian_group_structure","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab\\), the category of abelian groups"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab\\) — категория абелевых групп"}],"name":"abelian_groups_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category/arg/ab","labels":[],"name":"ab","type":{"name":"stacks.category_theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"name":"abelian_presheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0070: \\(F\\) is an abelian sheaf on \\(X\\) — an abelian presheaf whose underlying presheaf of sets is a sheaf"},{"language":"ru","status":"unofficial","text":"0070: \\(F\\) — абелев пучок на \\(X\\): абелев предпучок, чей предпучок множеств — пучок"}],"name":"abelian_sheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, existence: for every open covering \\(U = ∪ U_i\\) and sections \\(s_i ∈ F(U_i)\\) with \\(s_i|U_i∩U_j = s_j|U_i∩U_j\\) there exists \\(s ∈ F(U)\\) with \\(s|U_i = s_i\\)"},{"language":"ru","status":"unofficial","text":"006T, существование: для всякого открытого покрытия \\(U = ∪ U_i\\) и сечений \\(s_i ∈ F(U_i)\\), согласованных на пересечениях, существует \\(s ∈ F(U)\\) с \\(s|U_i = s_i\\)"}],"name":"compatible_sections_glue","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#flasque","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"09SW: the sheaf is flasque (flabby)"},{"language":"ru","status":"unofficial","text":"09SW: пучок вялый (flasque)"}],"name":"flasque","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#flasque/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"},{"language":"ru","status":"unofficial","text":"функтор есть \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"}],"name":"global_sections_functor","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Functor"}},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, uniqueness: a section \\(s ∈ F(U)\\) is determined by its restrictions \\(s|U_i\\) to an open covering"},{"language":"ru","status":"unofficial","text":"006T, единственность: сечение \\(s ∈ F(U)\\) определяется своими ограничениями \\(s|U_i\\) на открытое покрытие"}],"name":"gluing_is_unique","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)"},{"language":"ru","status":"unofficial","text":"0FKS: у пучка есть функториальная резольвента вялыми пучками (резольвента Годемана)"}],"name":"has_flasque_resolution","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is not a sheaf on \\(X\\): the sheaf condition 006T fails on some open covering"},{"language":"ru","status":"unofficial","text":"\\(F\\) — не пучок на \\(X\\): условие 006T провалено на некотором открытом покрытии"}],"name":"not_a_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a presheaf of sets on \\(X\\): a rule assigning a set \\(F(U)\\) to each open \\(U\\) and restriction maps \\(ρ^U_V\\) to inclusions \\(V ⊂ U\\) with \\(ρ^U_U = id\\) and \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — предпучок множеств на \\(X\\): правило, сопоставляющее каждому открытому \\(U\\) множество \\(F(U)\\), а вложениям \\(V ⊂ U\\) — отображения ограничения \\(ρ^U_V\\) с \\(ρ^U_U = id\\) и \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"}],"name":"presheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — пучок множеств на \\(X\\)"}],"name":"sheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab(X)\\), the category of sheaves of abelian groups on \\(X\\)"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab(X)\\) — категория пучков абелевых групп на \\(X\\)"}],"name":"sheaves_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/c","labels":[],"name":"c","type":{"name":"stacks.category_theory#Category"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_09SX","urn:stacks:clir:sheaf-cohomology#ST_09SY","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"},{"contributed":true,"fragmentCount":11,"fragments":["urn:stacks:clir:categories#ST_0109","urn:stacks:clir:categories#ST_010N","urn:stacks:clir:categories#ST_0135"],"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"}],"caseHash":"sha256:bbbb3a110161e09d0f7a6e2793f2b776e0dfbb7278bbd7c430796e7468451fde","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:3a0bacb658a20b29b5d6e7bd7e3fb02f592fc0d840ab878384bb7cfe11ae5aba","resultHash":"sha256:d587a046da0ef9a29dc3ed4d1f6bc4885af175d52f25c2a1767aa7fa1b404792","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:sh:f"],"facts":[{"args":["urn:case:stacks:sh:abx","urn:case:stacks:sh:x"],"predicate":"sheaves_category"},{"args":["urn:case:stacks:sh:ab"],"predicate":"abelian_groups_category"},{"args":["urn:case:stacks:sh:gamma","urn:case:stacks:sh:x"],"predicate":"global_sections_functor"},{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"predicate":"presheaf_of_sets_on"},{"args":["urn:case:stacks:sh:f"],"predicate":"abelian_group_structure"},{"args":["urn:case:stacks:sh:f"],"predicate":"compatible_sections_glue"},{"args":["urn:case:stacks:sh:f"],"predicate":"gluing_is_unique"},{"args":["urn:case:stacks:sh:f"],"package":"stacks-categories","predicate":"lifting_property"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"flasque","proof":true},"sources":[{"contentHash":"sha256:fa7c3dec36920848465ed2902237f8efade0877767eebcee37676bb1bb8717d9","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006K","kind":"fragment","locator":"tag/006K","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:b8388b54fb29de485ebd7c91f38c607f6005d41b4d427265a75b163118fd70d6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-presheaves}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf of abelian groups on $X$} or an\n{\\it abelian presheaf over $X$}\nis a presheaf of sets $\\mathcal{F}$ such that for each open\n$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with\nthe structure of an abelian group, and such that all restriction\nmaps $\\rho^U_V$ are homomorphisms of abelian groups, see\nLemma \\ref{lemma-abelian-presheaves} above.\n\\item A {\\it morphism of abelian presheaves over $X$}\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves\nof sets which induces\na homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nfor every open $U \\subset X$.\n\\item The category of presheaves of abelian groups on $X$ is denoted\n$\\textit{PAb}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:8504f30501fb40b30a60685a51aad76a416258206ec9a839891f7bf83cd38323","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006T","kind":"fragment","locator":"tag/006T","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:04f51a87121abeee45cd4e99f1379db299b359b4ac801017c7770a7f05d2c370","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_0070","kind":"fragment","locator":"tag/0070","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3260f7d5e9cbfe266160539b8375cd23ac2429762aaa12eb860ad707dc28a6cd","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item An {\\it abelian sheaf on $X$} or\n{\\it sheaf of abelian groups on $X$}\nis an abelian presheaf on $X$ such that the underlying presheaf of\nsets is a sheaf.\n\\item The category of sheaves of abelian groups\nis denoted $\\textit{Ab}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:d0262bf7656850a1933eef167a360437aa2bb6436e57461a3d23de09e8cc8ef3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01AD","kind":"fragment","locator":"tag/01AD","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf","language":"en","status":"official","text":"\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we work out basic notions of sheaves of modules.\nThis in particular includes the case of abelian sheaves, since\nthese may be viewed as sheaves of $\\underline{\\mathbf{Z}}$-modules.\nBasic references are \\cite{FAC}, \\cite{EGA} and \\cite{SGA4}.\n\n\\medskip\\noindent\nWe work out what happens for sheaves of modules on ringed topoi\nin another chapter (see\nModules on Sites, Section \\ref{sites-modules-section-introduction}),\nalthough there we will mostly just duplicate the discussion\nfrom this chapter."}]},{"contentHash":"sha256:1bf6bd6db42698470b675022b39392436590d133c6e4e0e6f342e63ce50e5290","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01AG","kind":"fragment","locator":"tag/01AG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:d40ca7777a52e10948b9c7e8da68432632fc0b8f4bb33c427370acaf48a4b19f","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover\na complex\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nis exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the\ncomplex\n$$\n\\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x\n$$\nis exact at $\\mathcal{G}_x$.\n\\end{lemma}"}]},{"contentHash":"sha256:48de83780b89577a36cf370ecf47370fa86fe2d17276908d3fb411375d5e1ec7","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_INJECTIVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01DG","kind":"fragment","locator":"tag/01DG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e4b4f09acb0a39efab590f7a0a1974dde000a27f6205d5284eb9b41b91b7f6d6","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian-sheaves-space}\nLet $X$ be a topological space.\nThe category of abelian sheaves on $X$ has enough injectives.\nIn fact it has functorial injective embeddings.\n\\end{lemma}"}]},{"contentHash":"sha256:0eee6ff9b997a29e65a9204cd10ad0d6c3337360eac630526713e3e18f1c9b03","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_0716","kind":"fragment","locator":"tag/0716","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:f61344b89b5447a2349b548070137ca4e6f491c90033bf637527fbd9f371c876","language":"en","status":"official","text":"\\section{Derived functors}\n\\label{section-derived-functors}\n\n\\noindent\nWe briefly explain how to get right derived functors using resolution\nfunctors. For the unbounded derived functors, please see\nSection \\ref{section-unbounded}.\n\n\\medskip\\noindent\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is abelian, see\nModules, Lemma \\ref{modules-lemma-abelian}.\nIn this chapter we will write\n$$\nK(\\mathcal{O}_X) = K(\\textit{Mod}(\\mathcal{O}_X))\n\\quad\n\\text{and}\n\\quad\nD(\\mathcal{O}_X) = D(\\textit{Mod}(\\mathcal{O}_X)).\n$$\nand similarly for the bounded versions for the triangulated categories\nintroduced in\nDerived Categories, Definition \\ref{derived-definition-complexes-notation} and\nDefinition \\ref{derived-definition-unbounded-derived-category}.\nBy\nDerived Categories, Remark \\ref{derived-remark-big-abelian-category}\nthere exists a resolution functor\n$$\nj = j_X :\nK^{+}(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK^{+}(\\mathcal{I})\n$$\nwhere $\\mathcal{I}$ is the strictly full additive subcategory of\n$\\textit{Mod}(\\mathcal{O}_X)$ consisting of injective sheaves.\nFor any left exact functor\n$F : \\textit{Mod}(\\mathcal{O}_X) \\to \\mathcal{B}$\ninto any abelian category $\\mathcal{B}$ we will denote $RF$ the\nright derived functor described in\nDerived Categories, Section \\ref{derived-section-right-derived-functor}\nand constructed using the resolution functor $j_X$ just described:\n\\begin{equation}\n\\label{equation-RF}\nRF = F \\circ j_X' : D^{+}(X) \\longrightarrow D^{+}(\\mathcal{B})\n\\end{equation}\nsee\nDerived Categories, Lemma \\ref{derived-lemma-right-derived-functor}\nfor notation. Note that we may think of $RF$ as defined on\n$\\textit{Mod}(\\mathcal{O}_X)$,\n$\\text{Comp}^{+}(\\textit{Mod}(\\mathcal{O}_X))$,\n$K^{+}(X)$, or $D^{+}(X)$\ndepending on the situation. According to\nDerived Categories, Definition \\ref{derived-definition-higher-derived-functors}\nwe obtain the $i$th right derived functor\n\\begin{equation}\n\\label{equation-RFi}\nR^iF = H^i \\circ RF : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\mathcal{B}\n\\end{equation}\nso that $R^0F = F$ and $\\{R^iF, \\delta\\}_{i \\geq 0}$ is universal\n$\\delta$-functor, see\nDerived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}.\n\n\\medskip\\noindent\nHere are two special cases of this construction.\nGiven a ring $R$ we write $K(R) = K(\\text{Mod}_R)$ and\n$D(R) = D(\\text{Mod}_R)$ and similarly for bounded versions.\nFor any open $U \\subset X$ we have a left exact functor\n$\n\\Gamma(U, -) :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\text{Mod}_{\\mathcal{O}_X(U)}\n$\nwhich gives rise to\n\\begin{equation}\n\\label{equation-total-derived-cohomology}\nR\\Gamma(U, -) :\nD^{+}(X)\n\\longrightarrow\nD^{+}(\\mathcal{O}_X(U))\n\\end{equation}\nby the discussion above. We set $H^i(U, -) = R^i\\Gamma(U, -)$.\nIf $U = X$ we recover (\\ref{equation-cohomology-modules}).\nIf $f : X \\to Y$ is a morphism of ringed spaces, then we have\nthe left exact functor\n$\nf_* :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_Y)\n$\nwhich gives rise to the {\\it derived pushforward}\n\\begin{equation}\n\\label{equation-total-derived-direct-image}\nRf_* :\nD^{+}(X)\n\\longrightarrow\nD^{+}(Y)\n\\end{equation}\nThe $i$th cohomology sheaf of $Rf_*\\mathcal{F}^\\bullet$ is denoted\n$R^if_*\\mathcal{F}^\\bullet$ and called the $i$th {\\it higher direct image}\nin accordance with (\\ref{equation-higher-direct-image-modules}).\nThe two displayed functors above are exact functors\nof derived categories.\n\n\\medskip\\noindent\n{\\bf Abuse of notation:} When the functor $Rf_*$, or any other\nderived functor, is applied to a sheaf $\\mathcal{F}$ on $X$ or a complex\nof sheaves it is understood that $\\mathcal{F}$ has been replaced by a\nsuitable resolution of $\\mathcal{F}$. To facilitate this kind of\noperation we will say, given an object\n$\\mathcal{F}^\\bullet \\in D(\\mathcal{O}_X)$,\nthat a bounded below complex $\\mathcal{I}^\\bullet$ of injectives of\n$\\textit{Mod}(\\mathcal{O}_X)$\n{\\it represents $\\mathcal{F}^\\bullet$ in the derived category}\nif there exists a quasi-isomorphism\n$\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$. In the same vein the phrase\n``let $\\alpha : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be\na morphism of $D(\\mathcal{O}_X)$''\ndoes not mean that $\\alpha$ is represented by a\nmorphism of complexes. If we have an actual morphism of complexes we will\nsay so."}]},{"contentHash":"sha256:f5e5d7a52fa030ef160ac88525e779a4a589ba55a59c29a59b9055850534792c","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_09SX","kind":"fragment","locator":"tag/09SX","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:d2662eefd4a3574b5c328b30b8abcb92218c774fd5994f253a2e2e7e4820d036","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-injective-flasque}\nLet $(X, \\mathcal{O}_X)$ be a ringed space.\nThen any injective $\\mathcal{O}_X$-module is flasque.\n\\end{lemma}"}]},{"contentHash":"sha256:7662615771c020c4afb51a8478766a67d4a3bdf502078614c0096d548d27dc05","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_09SY","kind":"fragment","locator":"tag/09SY","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:707ef368c24b80d3d736e07759dff795af4faae92c0ece2f17999e27898ff0d2","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-flasque-acyclic}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. Any flasque $\\mathcal{O}_X$-module\nis acyclic for $R\\Gamma(X, -)$ as well as $R\\Gamma(U, -)$ for any\nopen $U$ of $X$.\n\\end{lemma}"}]},{"contentHash":"sha256:787503a0ec46db68e4e11e9bf5fa2fabae3304c806bc7e2f5be69ab1af6cc11f","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0FKS","kind":"fragment","locator":"tag/0FKS","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:8b044713268f8366ca6cc0212d88985ff0517eb2417f464d218f454e93c283fe","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-godement-resolution}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution\n$$\n0 \\to\n\\mathcal{F} \\to\nf_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots\n$$\nfunctorial in $\\mathcal{F}$ such that each term\n$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque\n$\\mathcal{O}_X$-module and such that for all $x \\in X$ the\nmap\n$$\n\\mathcal{F}_x[0] \\to \\Big(\n(f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots\n\\Big)\n$$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.\n\\end{lemma}"}]},{"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":"flasque: **TRUE_ONLY** — установлено\nВыведено правом: abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx); has_flasque_resolution(urn:case:stacks:sh:f); injective_object(urn:case:stacks:sh:f); flasque(urn:case:stacks:sh:f); right_acyclic_for(urn:case:stacks:sh:f, urn:case:stacks:sh:gamma)\n…и ещё 15 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: InjectiveByLifting, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXEnoughInjectives, AbXKernels, AbelianSheafByDefinition, AbelianSheafIsObjectOfAbX, FlasqueIsAcyclic, GlobalSectionsBetween, GlobalSectionsLeftExact, GodementResolutionExists, InjectiveSheafIsFlasque, SheafByGluing, abelian_presheaf_on/sufficient\nОтвет поражаем правилом «006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)» — оно отменило бы вывод, будь установлено: not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:3a0bacb658a2…)\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\nproof-граф: 36 узлов — поле evaluation готово для law_explain"},{"answer":{"answer":{"evaluationStatus":"COMPUTED","kind":"TRUTH","meaning":"установлено","missingInputs":[],"truthStatus":"TRUE_ONLY"},"closedEditionRules":[],"derived":["abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx)","has_flasque_resolution(urn:case:stacks:sh:f)"],"derivedOmitted":11,"evaluation":{"proofGraph":{"nodes":[{"attributes":{"assertion":"urn:mcp:case#fact-1"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-1","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#enough_injectives"},"evidence":[],"id":"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbXKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-2"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-2","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#additive_category"},"evidence":[],"id":"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbAdditive","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#coimage_to_image_isomorphism"},"evidence":[],"id":"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_cokernels"},"evidence":[],"id":"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbCokernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#has_all_kernels"},"evidence":[],"id":"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-2"],"rule":"urn:stacks:clir:sheaf-cohomology#AbKernels","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-3"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-3","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#functor_between"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-1","urn:proof:assert:urn:mcp:case#fact-2","urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"},"v3":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:categories#preserves_left_exactness"},"evidence":[],"id":"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-3"],"rule":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-4"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-5"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"definition":{"concept":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","mode":"exact","part":"sufficient"}},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"evidence":[],"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{"assertion":"urn:mcp:case#fact-6"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{"assertion":"urn:mcp:case#fact-7"},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"evidence":[],"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","premises":[],"sourceAnchors":[]},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"evidence":[],"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"evidence":[],"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:category-theory#object_of"},"evidence":[],"id":"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","urn:proof:assert:urn:mcp:case#fact-1"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},"v2":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"evidence":[],"id":"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5","kind":"rule_application","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b"],"rule":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData","requirementStatus":"NEITHER","status":"UNDETERMINED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","kind":"constraint_check","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"constraint":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","requirementStatus":"TRUE_ONLY","status":"SATISFIED","triggerStatus":"SATISFIED"},"evidence":[],"id":"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","kind":"constraint_check","premises":["urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"sourceAnchors":[],"substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}}},{"attributes":{},"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"truthStatus":"TRUE_ONLY"},"evidence":[],"id":"urn:proof:query:mcp","kind":"query_evaluation","premises":["urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5"],"sourceAnchors":[]}],"proofHash":"sha256:52d5e4d5307887511d553d90c649d2ba104a2d328d4a11ae988443326a59df5f","roots":["urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7","urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd","urn:proof:query:mcp"]},"resultHash":"sha256:55697df015ad5cf4e575a4cd370d1b6492ec701a53968894f701f407f19207f5","schemaVersion":"law.core.evaluation/0.2"},"evaluationStatus":"COMPUTED","issues":[],"judgmentRequests":[],"proofRef":"urn:proof:query:mcp","provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:cbb748b0310e7cb0123ebe15d336ee802e2029c961418f557f9e1c2265f5200a","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e","resultHash":"sha256:55697df015ad5cf4e575a4cd370d1b6492ec701a53968894f701f407f19207f5","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"rulesApplied":["urn:stacks:clir:sheaf-cohomology#AbAdditive","urn:stacks:clir:sheaf-cohomology#AbCoimageImage","urn:stacks:clir:sheaf-cohomology#AbCokernels","urn:stacks:clir:sheaf-cohomology#AbKernels","urn:stacks:clir:sheaf-cohomology#AbXAdditive","urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","urn:stacks:clir:sheaf-cohomology#AbXCokernels","urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","urn:stacks:clir:sheaf-cohomology#AbXKernels","urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","urn:stacks:clir:sheaf-cohomology#SheafByGluing","urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient"],"signature":{"constants":{},"parameters":[{"labels":[],"name":"f","type":{"name":"urn:stacks:clir:category-theory#Obj"}}],"predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution","schemaVersion":"law.answers.signature/0.1","types":{"urn:stacks:clir:category-theory#Obj":{"kind":"unknown"}},"vocab":{}},"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","missing":["not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"],"premises":[{"premise":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)","status":"NEITHER"}],"rule":"NotSheafByRefutingCovering"}],"whyNot":[]},"execution":{"evaluationBytes":"{\"conflicts\":[],\"issues\":[],\"manifest\":{\"artifactHash\":\"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99\",\"calendarSnapshot\":\"\",\"caseHash\":\"sha256:cbb748b0310e7cb0123ebe15d336ee802e2029c961418f557f9e1c2265f5200a\",\"decisionTime\":\"2026-09-06T12:00:00+05:00\",\"evidenceSnapshotHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"externalSnapshots\":{},\"id\":\"urn:manifest:oracle-1\",\"interpretations\":[],\"knowledgeTime\":\"2026-09-06T12:00:00+05:00\",\"legalTime\":\"2026-09-06\",\"lockfileHash\":\"sha256:0000000000000000000000000000000000000000000000000000000000000000\",\"mode\":\"audit\",\"policies\":{},\"programHash\":\"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e\",\"resolvedEditions\":{},\"semanticHash\":\"sha256:bd4a0bf7389e1b3fe781710e276a96ef0f2fd89c438dd80df40833ee0def9166\",\"semantics\":\"law.core/0.2\",\"theoryHash\":\"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e\",\"timezone\":\"Asia/Qyzylorda\"},\"positions\":[],\"proofGraph\":{\"nodes\":[{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-1\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaves_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-1\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#enough_injectives\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbXKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-2\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_groups_category\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-2\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#additive_category\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbAdditive\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#coimage_to_image_isomorphism\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCoimageImage\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_cokernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbCokernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#has_all_kernels\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-2\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbKernels\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-3\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#global_sections_functor\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-3\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#functor_between\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-1\",\"urn:proof:assert:urn:mcp:case#fact-2\",\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:ab\",\"kind\":\"entity_ref\"},\"v3\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:categories#preserves_left_exactness\"},\"evidence\":[],\"id\":\"urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-3\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:gamma\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-4\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-4\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-5\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_group_structure\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-5\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"definition\":{\"concept\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\",\"mode\":\"exact\",\"part\":\"sufficient\"}},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-6\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-6\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{\"assertion\":\"urn:mcp:case#fact-7\"},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#gluing_is_unique\"},\"evidence\":[],\"id\":\"urn:proof:assert:urn:mcp:case#fact-7\",\"kind\":\"assertion\",\"premises\":[],\"sourceAnchors\":[]},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#SheafByGluing\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15\",\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:category-theory#object_of\"},\"evidence\":[],\"id\":\"urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\",\"urn:proof:assert:urn:mcp:case#fact-1\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:abx\",\"kind\":\"entity_ref\"},\"v2\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\"},\"evidence\":[],\"id\":\"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5\",\"kind\":\"rule_application\",\"premises\":[\"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b\"],\"rule\":\"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists\",\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"requirementStatus\":\"NEITHER\",\"status\":\"UNDETERMINED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-6\",\"urn:proof:assert:urn:mcp:case#fact-7\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"constraint\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"requirementStatus\":\"TRUE_ONLY\",\"status\":\"SATISFIED\",\"triggerStatus\":\"SATISFIED\"},\"evidence\":[],\"id\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"kind\":\"constraint_check\",\"premises\":[\"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550\",\"urn:proof:assert:urn:mcp:case#fact-4\",\"urn:proof:assert:urn:mcp:case#fact-5\"],\"sourceAnchors\":[],\"substitution\":{\"v0\":{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"},\"v1\":{\"id\":\"urn:case:stacks:sh:x\",\"kind\":\"entity_ref\"}}},{\"attributes\":{},\"conclusion\":{\"literal\":{\"args\":[{\"id\":\"urn:case:stacks:sh:f\",\"kind\":\"entity_ref\"}],\"kind\":\"literal\",\"polarity\":\"positive\",\"predicate\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\"},\"truthStatus\":\"TRUE_ONLY\"},\"evidence\":[],\"id\":\"urn:proof:query:mcp\",\"kind\":\"query_evaluation\",\"premises\":[\"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5\"],\"sourceAnchors\":[]}],\"proofHash\":\"sha256:52d5e4d5307887511d553d90c649d2ba104a2d328d4a11ae988443326a59df5f\",\"roots\":[\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"urn:proof:query:mcp\"]},\"resultHash\":\"sha256:55697df015ad5cf4e575a4cd370d1b6492ec701a53968894f701f407f19207f5\",\"results\":[{\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:mcp\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatusSupports\":[],\"proof\":\"urn:proof:query:mcp\",\"query\":\"urn:query:mcp\",\"resultKind\":\"PROPOSITION\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution\",\"truthStatus\":\"TRUE_ONLY\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:c183ec1c564d2818\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[{\"description\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf: опора не установлена (§93.2)\",\"id\":\"urn:result:constraint:DeclaredSheafNotRefutedByData:c183ec1c564d2818:input:0\",\"kind\":\"fact\",\"relatedNode\":\"urn:stacks:clir:sheaf-cohomology#not_a_sheaf\"}],\"normativeStatus\":\"UNDETERMINED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#DeclaredSheafNotRefutedByData\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"NEITHER\"},{\"applicabilityStatus\":\"APPLICABLE\",\"conflicts\":[],\"evaluationStatus\":\"COMPUTED\",\"evidence\":[],\"id\":\"urn:result:constraint:abelian_presheaf_on/necessary:b945bb089fa6cd6e\",\"judgmentRequests\":[],\"manifest\":\"urn:manifest:oracle-1\",\"missingInputs\":[],\"normativeStatus\":\"SATISFIED\",\"normativeStatusSupports\":[],\"proof\":\"urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd\",\"query\":\"urn:query:mcp\",\"resultKind\":\"CONSTRAINT\",\"sourceAnchors\":[],\"target\":\"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary\",\"triggerStatus\":\"SATISFIED\",\"truthStatus\":\"TRUE_ONLY\"}],\"schemaVersion\":\"law.core.evaluation/0.2\"}","evaluationSha256":"sha256:22dd3ca08c675673eb855d605d37cb8e132c3aca3d914470b02f19fe7ff86530","request":{"case":{"assertions":[{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-1","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:abx","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaves_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-2","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:ab","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-3","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:gamma","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#global_sections_functor"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-4","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-5","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-6","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"},{"contentHash":"sha256:0000000000000000000000000000000000000000000000000000000000000000","evidence":[],"id":"urn:mcp:case#fact-7","kind":"assertion","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"origin":"case_input","package":"urn:stacks:clir:sheaf-cohomology"}],"context":{"decisionTime":"2026-09-06T12:00:00+05:00","knowledgeTime":"2026-09-06T12:00:00+05:00","legalTime":"2026-09-06","timezone":"Asia/Qyzylorda"},"options":{"selectedInterpretations":[]}},"ir":{"irSha256":"sha256:63cba186192426450399328e45e57b3ee07713867431d6f26c106fd5a6bae87e","kind":"world_ref","nodeCount":181,"packages":[{"artifactHash":"sha256:5269ca43c11ea65fffcd8e3589fba8c3a899f2d863c75d0992e8b0cf57a9ea99","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","semanticHash":"sha256:1ee2443e666f565715a3e3a5663914219d324c205aa6004f03c007d582687303"}],"programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e"},"query":{"kind":"truth","literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"queryId":"urn:query:mcp"},"schemaVersion":"law.core.evaluation-request/0.2","semanticVersion":"0.2"},"requestCanonicalSha256":"sha256:ffec8dcab12698dc6bca6fb20f24947de8be652e1f3cbac0c0fd8e3bc1fd1600"},"id":"godement","kind":"truth","label":"Вялая резольвента существует","presentation":{"blockers":{"vulnerableTo":[{"anchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"label":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)","premises":[{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"missing":true,"predicate":"not_a_sheaf","predicateId":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","status":"NEITHER","text":"not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)"}],"rule":"NotSheafByRefutingCovering","ruleId":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering"}]},"schemaVersion":"law.answers.presentation/0.1","source":{"requestCanonicalSha256":"sha256:ffec8dcab12698dc6bca6fb20f24947de8be652e1f3cbac0c0fd8e3bc1fd1600"},"steps":[{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on"},"id":"urn:proof:assert:urn:mcp:case#fact-4","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure"},"id":"urn:proof:assert:urn:mcp:case#fact-5","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on"},"id":"urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-5"],"rule":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006K"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue"},"id":"urn:proof:assert:urn:mcp:case#fact-6","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"assertionOrigin":"case_input","conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique"},"id":"urn:proof:assert:urn:mcp:case#fact-7","kind":"assertion","originStatus":"available","premises":[],"trust":"case"},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on"},"id":"urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","kind":"rule_application","originStatus":"available","premises":["urn:proof:assert:urn:mcp:case#fact-4","urn:proof:assert:urn:mcp:case#fact-6","urn:proof:assert:urn:mcp:case#fact-7"],"rule":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_006T"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"head":{"args":[{"kind":"var","var":"v0"},{"kind":"var","var":"v1"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on"},"id":"urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15","urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"],"rule":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0070"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"head":{"args":[{"kind":"var","var":"v0"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"id":"urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5","kind":"rule_application","originStatus":"available","premises":["urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b"],"rule":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","sourceAnchors":["urn:stacks:clir:sheaf-cohomology#ST_0FKS","urn:stacks:clir:sheaf-cohomology#ST_01AD"],"strength":"strict","substitution":{"v0":{"id":"urn:case:stacks:sh:f","kind":"entity_ref"},"v1":{"id":"urn:case:stacks:sh:x","kind":"entity_ref"}},"trust":"engine","variables":[{"id":"v0","type":{"name":"stacks.category_theory#Obj"}},{"id":"v1","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}]},{"conclusion":{"literal":{"args":[{"id":"urn:case:stacks:sh:f","kind":"entity_ref"}],"kind":"literal","polarity":"positive","predicate":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"},"truthStatus":"TRUE_ONLY"},"id":"urn:proof:query:mcp","kind":"query_evaluation","originStatus":"available","premises":["urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5"],"trust":"engine"}],"symbols":[{"id":"urn:stacks:clir:sheaf-cohomology#AbAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG on the one-point space: \\(Ab = Mod(Z)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG на одноточечном пространстве: \\(Ab = Mod(Z)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXAdditive","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular additive"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности аддитивна"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCoimageImage","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular \\(Coim(f) → Im(f)\\) is an isomorphism for every morphism"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности \\(Coim(f) → Im(f)\\) — изоморфизм для всякого морфизма"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXCokernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all cokernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все коядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives","kind":"rule","labels":[{"language":"en","status":"official","text":"01DG: the category of abelian sheaves on a topological space \\(X\\) has enough injectives"},{"language":"ru","status":"unofficial","text":"01DG: в категории абелевых пучков на топологическом пространстве \\(X\\) достаточно инъективных"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbXKernels","kind":"rule","labels":[{"language":"en","status":"official","text":"01AG with 01AD: \\(Ab(X) = Mod(Z_X)\\) is abelian, in particular all kernels exist"},{"language":"ru","status":"unofficial","text":"01AG с 01AD: \\(Ab(X) = Mod(Z_X)\\) абелева, в частности все ядра существуют"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (1): an abelian presheaf on \\(X\\) whose underlying presheaf of sets is a sheaf is an abelian sheaf"},{"language":"ru","status":"unofficial","text":"0070 (1): абелев предпучок на \\(X\\), чей предпучок множеств — пучок, есть абелев пучок"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX","kind":"rule","labels":[{"language":"en","status":"official","text":"0070 (2): an abelian sheaf on \\(X\\) is an object of \\(Ab(X)\\)"},{"language":"ru","status":"unofficial","text":"0070 (2): абелев пучок на \\(X\\) — объект категории \\(Ab(X)\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a functor from \\(Ab(X)\\) to \\(Ab\\)"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — функтор из \\(Ab(X)\\) в \\(Ab\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact","kind":"rule","labels":[{"language":"en","status":"official","text":"0716: \\(Γ(X, −)\\) is a left exact functor"},{"language":"ru","status":"unofficial","text":"0716: \\(Γ(X, −)\\) — точный слева функтор"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#GodementResolutionExists","kind":"rule","labels":[{"language":"en","status":"official","text":"0FKS with 01AD: every abelian sheaf \\(F\\) on \\(X\\) has the Godement resolution \\(0 → F → f_*f^*F → …\\) by flasque sheaves"},{"language":"ru","status":"unofficial","text":"0FKS с 01AD: у всякого абелева пучка \\(F\\) на \\(X\\) есть резольвента Годемана \\(0 → F → f_*f^*F → …\\) вялыми пучками"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"006T: предпучок, опровергнутый открытым покрытием, — не пучок множеств на \\(X\\)"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#SheafByGluing","kind":"rule","labels":[{"language":"en","status":"official","text":"006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section"},{"language":"ru","status":"unofficial","text":"006T: предпучок множеств — пучок, если согласованные семейства сечений над любым открытым покрытием склеиваются в единственное сечение"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#Space","kind":"type_decl","labels":[{"language":"en","status":"official","text":"topological space"},{"language":"ru","status":"unofficial","text":"топологическое пространство"}],"name":"Space","package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"each \\(F(U)\\) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups"},{"language":"ru","status":"unofficial","text":"каждое \\(F(U)\\) несёт структуру абелевой группы, и все отображения ограничения — гомоморфизмы абелевых групп"}],"name":"abelian_group_structure","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_group_structure/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab\\), the category of abelian groups"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab\\) — категория абелевых групп"}],"name":"abelian_groups_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_groups_category/arg/ab","labels":[],"name":"ab","type":{"name":"stacks.category_theory#Category"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"name":"abelian_presheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/necessary","kind":"constraint","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient","kind":"rule","labels":[{"language":"en","status":"official","text":"006K: an abelian presheaf on \\(X\\) is a presheaf of sets \\(F\\) such that each \\(F(U)\\) is an abelian group and all restriction maps are group homomorphisms"},{"language":"ru","status":"unofficial","text":"006K: абелев предпучок на \\(X\\) — предпучок множеств \\(F\\), у которого каждое \\(F(U)\\) — абелева группа, а все ограничения — гомоморфизмы групп"}],"package":"urn:stacks:clir:sheaf-cohomology","strength":"strict"},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0070: \\(F\\) is an abelian sheaf on \\(X\\) — an abelian presheaf whose underlying presheaf of sets is a sheaf"},{"language":"ru","status":"unofficial","text":"0070: \\(F\\) — абелев пучок на \\(X\\): абелев предпучок, чей предпучок множеств — пучок"}],"name":"abelian_sheaf_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#abelian_sheaf_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, existence: for every open covering \\(U = ∪ U_i\\) and sections \\(s_i ∈ F(U_i)\\) with \\(s_i|U_i∩U_j = s_j|U_i∩U_j\\) there exists \\(s ∈ F(U)\\) with \\(s|U_i = s_i\\)"},{"language":"ru","status":"unofficial","text":"006T, существование: для всякого открытого покрытия \\(U = ∪ U_i\\) и сечений \\(s_i ∈ F(U_i)\\), согласованных на пересечениях, существует \\(s ∈ F(U)\\) с \\(s|U_i = s_i\\)"}],"name":"compatible_sections_glue","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#compatible_sections_glue/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the functor is \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"},{"language":"ru","status":"unofficial","text":"функтор есть \\(Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)\\)"}],"name":"global_sections_functor","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/g","labels":[],"name":"g","type":{"name":"stacks.category_theory#Functor"}},{"id":"urn:stacks:clir:sheaf-cohomology#global_sections_functor/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"006T, uniqueness: a section \\(s ∈ F(U)\\) is determined by its restrictions \\(s|U_i\\) to an open covering"},{"language":"ru","status":"unofficial","text":"006T, единственность: сечение \\(s ∈ F(U)\\) определяется своими ограничениями \\(s|U_i\\) на открытое покрытие"}],"name":"gluing_is_unique","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#gluing_is_unique/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)"},{"language":"ru","status":"unofficial","text":"0FKS: у пучка есть функториальная резольвента вялыми пучками (резольвента Годемана)"}],"name":"has_flasque_resolution","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#has_flasque_resolution/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is not a sheaf on \\(X\\): the sheaf condition 006T fails on some open covering"},{"language":"ru","status":"unofficial","text":"\\(F\\) — не пучок на \\(X\\): условие 006T провалено на некотором открытом покрытии"}],"name":"not_a_sheaf","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#not_a_sheaf/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a presheaf of sets on \\(X\\): a rule assigning a set \\(F(U)\\) to each open \\(U\\) and restriction maps \\(ρ^U_V\\) to inclusions \\(V ⊂ U\\) with \\(ρ^U_U = id\\) and \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — предпучок множеств на \\(X\\): правило, сопоставляющее каждому открытому \\(U\\) множество \\(F(U)\\), а вложениям \\(V ⊂ U\\) — отображения ограничения \\(ρ^U_V\\) с \\(ρ^U_U = id\\) и \\(ρ^U_W = ρ^V_W ∘ ρ^U_V\\)"}],"name":"presheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#presheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"\\(F\\) is a sheaf of sets on \\(X\\)"},{"language":"ru","status":"unofficial","text":"\\(F\\) — пучок множеств на \\(X\\)"}],"name":"sheaf_of_sets_on","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/f","labels":[],"name":"f","type":{"name":"stacks.category_theory#Obj"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaf_of_sets_on/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category","kind":"symbol_decl","labels":[{"language":"en","status":"official","text":"the category is \\(Ab(X)\\), the category of sheaves of abelian groups on \\(X\\)"},{"language":"ru","status":"unofficial","text":"категория есть \\(Ab(X)\\) — категория пучков абелевых групп на \\(X\\)"}],"name":"sheaves_category","package":"urn:stacks:clir:sheaf-cohomology","parameters":[{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/c","labels":[],"name":"c","type":{"name":"stacks.category_theory#Category"}},{"id":"urn:stacks:clir:sheaf-cohomology#sheaves_category/arg/x","labels":[],"name":"x","type":{"name":"urn:stacks:clir:sheaf-cohomology#Space"}}],"symbolKind":"relation"}],"tables":[]},"provenance":{"acts":[{"contributed":true,"fragmentCount":26,"fragments":["urn:stacks:clir:sheaf-cohomology#ST_006K","urn:stacks:clir:sheaf-cohomology#ST_006T","urn:stacks:clir:sheaf-cohomology#ST_0070","urn:stacks:clir:sheaf-cohomology#ST_01AD","urn:stacks:clir:sheaf-cohomology#ST_01AG","urn:stacks:clir:sheaf-cohomology#ST_01DG","urn:stacks:clir:sheaf-cohomology#ST_0716","urn:stacks:clir:sheaf-cohomology#ST_0FKS"],"jurisdiction":"none","namespace":"urn:stacks:clir:sheaf-cohomology","package":"stacks-sheaf-cohomology","title":"Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"}],"caseHash":"sha256:cbb748b0310e7cb0123ebe15d336ee802e2029c961418f557f9e1c2265f5200a","codeHash":"sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699","jurisdiction":"вне юрисдикции государства","legalTime":"2026-09-06","mode":"audit","programHash":"sha256:6b62eb59e903172fce7cc3ce1a8e29b1fbfc4c6808acc817493c98a07cd7e74e","resultHash":"sha256:55697df015ad5cf4e575a4cd370d1b6492ec701a53968894f701f407f19207f5","rustCodeHash":"sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4","timezone":"Asia/Qyzylorda"},"request":{"args":["urn:case:stacks:sh:f"],"facts":[{"args":["urn:case:stacks:sh:abx","urn:case:stacks:sh:x"],"predicate":"sheaves_category"},{"args":["urn:case:stacks:sh:ab"],"predicate":"abelian_groups_category"},{"args":["urn:case:stacks:sh:gamma","urn:case:stacks:sh:x"],"predicate":"global_sections_functor"},{"args":["urn:case:stacks:sh:f","urn:case:stacks:sh:x"],"predicate":"presheaf_of_sets_on"},{"args":["urn:case:stacks:sh:f"],"predicate":"abelian_group_structure"},{"args":["urn:case:stacks:sh:f"],"predicate":"compatible_sections_glue"},{"args":["urn:case:stacks:sh:f"],"predicate":"gluing_is_unique"}],"kind":"truth","legalTime":"2026-09-06","package":"stacks-sheaf-cohomology","predicate":"has_flasque_resolution","proof":true},"sources":[{"contentHash":"sha256:fa7c3dec36920848465ed2902237f8efade0877767eebcee37676bb1bb8717d9","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006K","kind":"fragment","locator":"tag/006K","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:b8388b54fb29de485ebd7c91f38c607f6005d41b4d427265a75b163118fd70d6","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-presheaves}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it presheaf of abelian groups on $X$} or an\n{\\it abelian presheaf over $X$}\nis a presheaf of sets $\\mathcal{F}$ such that for each open\n$U \\subset X$ the set $\\mathcal{F}(U)$ is endowed with\nthe structure of an abelian group, and such that all restriction\nmaps $\\rho^U_V$ are homomorphisms of abelian groups, see\nLemma \\ref{lemma-abelian-presheaves} above.\n\\item A {\\it morphism of abelian presheaves over $X$}\n$\\varphi : \\mathcal{F} \\to \\mathcal{G}$ is a morphism of presheaves\nof sets which induces\na homomorphism of abelian groups $\\mathcal{F}(U) \\to \\mathcal{G}(U)$\nfor every open $U \\subset X$.\n\\item The category of presheaves of abelian groups on $X$ is denoted\n$\\textit{PAb}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:8504f30501fb40b30a60685a51aad76a416258206ec9a839891f7bf83cd38323","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_006T","kind":"fragment","locator":"tag/006T","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:041460b360dfa6a9d137bed749118fd98c2764c6ac5b9058dcb5aa7167313f76","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item A {\\it sheaf $\\mathcal{F}$ of sets on $X$} is a presheaf\nof sets which satisfies the following additional property: Given\nany open covering $U = \\bigcup_{i \\in I} U_i$ and any collection\nof sections $s_i \\in \\mathcal{F}(U_i)$, $i \\in I$ such that\n$\\forall i, j\\in I$\n$$\ns_i|_{U_i \\cap U_j} = s_j|_{U_i \\cap U_j}\n$$\nthere exists a unique section $s \\in \\mathcal{F}(U)$ such that\n$s_i = s|_{U_i}$ for all $i \\in I$.\n\\item A {\\it morphism of sheaves of sets} is simply a\nmorphism of presheaves of sets.\n\\item The category of sheaves of sets on $X$ is denoted\n$\\Sh(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:04f51a87121abeee45cd4e99f1379db299b359b4ac801017c7770a7f05d2c370","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER","fragmentKind":"defn","id":"urn:stacks:clir:sheaf-cohomology#ST_0070","kind":"fragment","locator":"tag/0070","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3260f7d5e9cbfe266160539b8375cd23ac2429762aaa12eb860ad707dc28a6cd","language":"en","status":"official","text":"\\begin{definition}\n\\label{definition-abelian-sheaf}\nLet $X$ be a topological space.\n\\begin{enumerate}\n\\item An {\\it abelian sheaf on $X$} or\n{\\it sheaf of abelian groups on $X$}\nis an abelian presheaf on $X$ such that the underlying presheaf of\nsets is a sheaf.\n\\item The category of sheaves of abelian groups\nis denoted $\\textit{Ab}(X)$.\n\\end{enumerate}\n\\end{definition}"}]},{"contentHash":"sha256:d0262bf7656850a1933eef167a360437aa2bb6436e57461a3d23de09e8cc8ef3","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_01AD","kind":"fragment","locator":"tag/01AD","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf","language":"en","status":"official","text":"\\section{Introduction}\n\\label{section-introduction}\n\n\\noindent\nIn this chapter we work out basic notions of sheaves of modules.\nThis in particular includes the case of abelian sheaves, since\nthese may be viewed as sheaves of $\\underline{\\mathbf{Z}}$-modules.\nBasic references are \\cite{FAC}, \\cite{EGA} and \\cite{SGA4}.\n\n\\medskip\\noindent\nWe work out what happens for sheaves of modules on ringed topoi\nin another chapter (see\nModules on Sites, Section \\ref{sites-modules-section-introduction}),\nalthough there we will mostly just duplicate the discussion\nfrom this chapter."}]},{"contentHash":"sha256:1bf6bd6db42698470b675022b39392436590d133c6e4e0e6f342e63ce50e5290","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01AG","kind":"fragment","locator":"tag/01AG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:d40ca7777a52e10948b9c7e8da68432632fc0b8f4bb33c427370acaf48a4b19f","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is an abelian category. Moreover\na complex\n$$\n\\mathcal{F} \\to \\mathcal{G} \\to \\mathcal{H}\n$$\nis exact at $\\mathcal{G}$ if and only if for all $x \\in X$ the\ncomplex\n$$\n\\mathcal{F}_x \\to \\mathcal{G}_x \\to \\mathcal{H}_x\n$$\nis exact at $\\mathcal{G}_x$.\n\\end{lemma}"}]},{"contentHash":"sha256:48de83780b89577a36cf370ecf47370fa86fe2d17276908d3fb411375d5e1ec7","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_INJECTIVES_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_01DG","kind":"fragment","locator":"tag/01DG","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:e4b4f09acb0a39efab590f7a0a1974dde000a27f6205d5284eb9b41b91b7f6d6","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-abelian-sheaves-space}\nLet $X$ be a topological space.\nThe category of abelian sheaves on $X$ has enough injectives.\nIn fact it has functorial injective embeddings.\n\\end{lemma}"}]},{"contentHash":"sha256:0eee6ff9b997a29e65a9204cd10ad0d6c3337360eac630526713e3e18f1c9b03","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"section","id":"urn:stacks:clir:sheaf-cohomology#ST_0716","kind":"fragment","locator":"tag/0716","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:f61344b89b5447a2349b548070137ca4e6f491c90033bf637527fbd9f371c876","language":"en","status":"official","text":"\\section{Derived functors}\n\\label{section-derived-functors}\n\n\\noindent\nWe briefly explain how to get right derived functors using resolution\nfunctors. For the unbounded derived functors, please see\nSection \\ref{section-unbounded}.\n\n\\medskip\\noindent\nLet $(X, \\mathcal{O}_X)$ be a ringed space. The category\n$\\textit{Mod}(\\mathcal{O}_X)$ is abelian, see\nModules, Lemma \\ref{modules-lemma-abelian}.\nIn this chapter we will write\n$$\nK(\\mathcal{O}_X) = K(\\textit{Mod}(\\mathcal{O}_X))\n\\quad\n\\text{and}\n\\quad\nD(\\mathcal{O}_X) = D(\\textit{Mod}(\\mathcal{O}_X)).\n$$\nand similarly for the bounded versions for the triangulated categories\nintroduced in\nDerived Categories, Definition \\ref{derived-definition-complexes-notation} and\nDefinition \\ref{derived-definition-unbounded-derived-category}.\nBy\nDerived Categories, Remark \\ref{derived-remark-big-abelian-category}\nthere exists a resolution functor\n$$\nj = j_X :\nK^{+}(\\textit{Mod}(\\mathcal{O}_X))\n\\longrightarrow\nK^{+}(\\mathcal{I})\n$$\nwhere $\\mathcal{I}$ is the strictly full additive subcategory of\n$\\textit{Mod}(\\mathcal{O}_X)$ consisting of injective sheaves.\nFor any left exact functor\n$F : \\textit{Mod}(\\mathcal{O}_X) \\to \\mathcal{B}$\ninto any abelian category $\\mathcal{B}$ we will denote $RF$ the\nright derived functor described in\nDerived Categories, Section \\ref{derived-section-right-derived-functor}\nand constructed using the resolution functor $j_X$ just described:\n\\begin{equation}\n\\label{equation-RF}\nRF = F \\circ j_X' : D^{+}(X) \\longrightarrow D^{+}(\\mathcal{B})\n\\end{equation}\nsee\nDerived Categories, Lemma \\ref{derived-lemma-right-derived-functor}\nfor notation. Note that we may think of $RF$ as defined on\n$\\textit{Mod}(\\mathcal{O}_X)$,\n$\\text{Comp}^{+}(\\textit{Mod}(\\mathcal{O}_X))$,\n$K^{+}(X)$, or $D^{+}(X)$\ndepending on the situation. According to\nDerived Categories, Definition \\ref{derived-definition-higher-derived-functors}\nwe obtain the $i$th right derived functor\n\\begin{equation}\n\\label{equation-RFi}\nR^iF = H^i \\circ RF : \\textit{Mod}(\\mathcal{O}_X) \\longrightarrow \\mathcal{B}\n\\end{equation}\nso that $R^0F = F$ and $\\{R^iF, \\delta\\}_{i \\geq 0}$ is universal\n$\\delta$-functor, see\nDerived Categories, Lemma \\ref{derived-lemma-higher-derived-functors}.\n\n\\medskip\\noindent\nHere are two special cases of this construction.\nGiven a ring $R$ we write $K(R) = K(\\text{Mod}_R)$ and\n$D(R) = D(\\text{Mod}_R)$ and similarly for bounded versions.\nFor any open $U \\subset X$ we have a left exact functor\n$\n\\Gamma(U, -) :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\text{Mod}_{\\mathcal{O}_X(U)}\n$\nwhich gives rise to\n\\begin{equation}\n\\label{equation-total-derived-cohomology}\nR\\Gamma(U, -) :\nD^{+}(X)\n\\longrightarrow\nD^{+}(\\mathcal{O}_X(U))\n\\end{equation}\nby the discussion above. We set $H^i(U, -) = R^i\\Gamma(U, -)$.\nIf $U = X$ we recover (\\ref{equation-cohomology-modules}).\nIf $f : X \\to Y$ is a morphism of ringed spaces, then we have\nthe left exact functor\n$\nf_* :\n\\textit{Mod}(\\mathcal{O}_X)\n\\longrightarrow\n\\textit{Mod}(\\mathcal{O}_Y)\n$\nwhich gives rise to the {\\it derived pushforward}\n\\begin{equation}\n\\label{equation-total-derived-direct-image}\nRf_* :\nD^{+}(X)\n\\longrightarrow\nD^{+}(Y)\n\\end{equation}\nThe $i$th cohomology sheaf of $Rf_*\\mathcal{F}^\\bullet$ is denoted\n$R^if_*\\mathcal{F}^\\bullet$ and called the $i$th {\\it higher direct image}\nin accordance with (\\ref{equation-higher-direct-image-modules}).\nThe two displayed functors above are exact functors\nof derived categories.\n\n\\medskip\\noindent\n{\\bf Abuse of notation:} When the functor $Rf_*$, or any other\nderived functor, is applied to a sheaf $\\mathcal{F}$ on $X$ or a complex\nof sheaves it is understood that $\\mathcal{F}$ has been replaced by a\nsuitable resolution of $\\mathcal{F}$. To facilitate this kind of\noperation we will say, given an object\n$\\mathcal{F}^\\bullet \\in D(\\mathcal{O}_X)$,\nthat a bounded below complex $\\mathcal{I}^\\bullet$ of injectives of\n$\\textit{Mod}(\\mathcal{O}_X)$\n{\\it represents $\\mathcal{F}^\\bullet$ in the derived category}\nif there exists a quasi-isomorphism\n$\\mathcal{F}^\\bullet \\to \\mathcal{I}^\\bullet$. In the same vein the phrase\n``let $\\alpha : \\mathcal{F}^\\bullet \\to \\mathcal{G}^\\bullet$ be\na morphism of $D(\\mathcal{O}_X)$''\ndoes not mean that $\\alpha$ is represented by a\nmorphism of complexes. If we have an actual morphism of complexes we will\nsay so."}]},{"contentHash":"sha256:787503a0ec46db68e4e11e9bf5fa2fabae3304c806bc7e2f5be69ab1af6cc11f","edition":"urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER","fragmentKind":"lemma","id":"urn:stacks:clir:sheaf-cohomology#ST_0FKS","kind":"fragment","locator":"tag/0FKS","package":"urn:stacks:clir:sheaf-cohomology","texts":[{"contentHash":"sha256:8b044713268f8366ca6cc0212d88985ff0517eb2417f464d218f454e93c283fe","language":"en","status":"official","text":"\\begin{lemma}\n\\label{lemma-godement-resolution}\nLet $(X, \\mathcal{O}_X)$ be a ringed space. For every sheaf of\n$\\mathcal{O}_X$-modules $\\mathcal{F}$ there is a resolution\n$$\n0 \\to\n\\mathcal{F} \\to\nf_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*\\mathcal{F} \\to\nf_*f^*f_*f^*f_*f^*\\mathcal{F} \\to \\ldots\n$$\nfunctorial in $\\mathcal{F}$ such that each term\n$f_*f^* \\ldots f_*f^*\\mathcal{F}$ is a flasque\n$\\mathcal{O}_X$-module and such that for all $x \\in X$ the\nmap\n$$\n\\mathcal{F}_x[0] \\to \\Big(\n(f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*\\mathcal{F})_x \\to\n(f_*f^*f_*f^*f_*f^*\\mathcal{F})_x \\to \\ldots\n\\Big)\n$$\nis a homotopy equivalence in the category of complexes\nof $\\mathcal{O}_{X, x}$-modules.\n\\end{lemma}"}]}],"text":"has_flasque_resolution: **TRUE_ONLY** — установлено\nВыведено правом: abelian_presheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); sheaf_of_sets_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x); object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx); has_flasque_resolution(urn:case:stacks:sh:f)\n…и ещё 11 выведенных фактов вне предмета вопроса (полный вывод — law_explain)\nПрименены правила: AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXEnoughInjectives, AbXKernels, AbelianSheafByDefinition, AbelianSheafIsObjectOfAbX, GlobalSectionsBetween, GlobalSectionsLeftExact, GodementResolutionExists, SheafByGluing, abelian_presheaf_on/sufficient\nОтвет поражаем правилом «006T: a presheaf refuted by an open covering is not a sheaf of sets on \\(X\\)» — оно отменило бы вывод, будь установлено: not_a_sheaf(urn:case:stacks:sh:f, urn:case:stacks:sh:x)\n(поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле)\nПраво (вне юрисдикции государства): Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — доктрина (programHash sha256:6b62eb59e903…)\nproof-граф: 26 узлов — поле evaluation готово для law_explain"}],"language":"ru","question":{"origin":"user","text":"Почему H⁰(X, F) — это в точности глобальные сечения, отчего вялый пучок не имеет старших когомологий и всякий ли пучок допускает вялую резольвенту?"},"schemaVersion":"law.answers.document/0.1"}