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Почему H⁰(X, F) — это в точности глобальные сечения, отчего вялый пучок не имеет старших когомологий и всякий ли пучок допускает вялую резольвенту?

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.

This is an assistant explanation, not a calculation result. Check the grounds and sources below.

At a glance

6

Select a result to explore its grounds

Пучкование даёт пучок абелевых группConditionEstablishedExecution: ComputedLogical result: EstablishedУниверсальность пучкованияConditionEstablishedExecution: ComputedLogical result: EstablishedH⁰ равна глобальным сечениямConditionEstablishedExecution: ComputedLogical result: EstablishedСтаршие когомологии вялого пучка нулевыеConditionEstablishedExecution: ComputedLogical result: EstablishedИнъективный пучок вялConditionEstablishedExecution: ComputedLogical result: EstablishedВялая резольвента существуетConditionEstablishedExecution: ComputedLogical result: Established

Detailed analysis

6

Condition

Пучкование даёт пучок абелевых групп

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf

fsharpx

Input facts

  • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: fx: x

  • each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: f

  • G=F#G = F^# is the sheafification of the presheaf FF : sections over UU are compatible families of germs (Sheaves, Section 007X), with the canonical map F→F#F → F^#

    g: fsharpf: f

Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "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
}
Why this resultApplied rules and conditions

Derivation path6 steps

  1. 1

    FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    case fact
  2. 2

    each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: urn:case:stacks:sh:f

    case fact
  3. 3

    006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006K

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
    rule
  4. 4

    G=F#G = F^# is the sheafification of the presheaf FF : sections over UU are compatible families of germs (Sheaves, Section 007X), with the canonical map F→F#F → F^#

    g: urn:case:stacks:sh:fsharp; f: urn:case:stacks:sh:f

    case fact
  5. 5

    0085: for an abelian presheaf FF there is a unique abelian sheaf structure on F#F^# making F→F#F → F^# a morphism of abelian presheaves

    0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf: f: urn:case:stacks:sh:fsharp; x: urn:case:stacks:sh:x

    tag 0085

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf
    rule
  6. 6

    Query evaluation

    query

verified by the engine: 3 · case fact: 3 · Full graph: 10 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0085: for an abelian presheaf FF there is a unique abelian sheaf structure on F#F^# making F→F#F → F^# a morphism of abelian presheaves

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafifyAbelianPresheaf

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
Other rules in the evaluation3

Applied in the overall evaluation, but not on the proof path for this answer.

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0FKS with 01AD: every abelian sheaf FF on XX has the Godement resolution 0→F→f*f*F→…0 → F → f_*f^*F → … by flasque sheaves

    Identifier
    urn:stacks:clir:sheaf-cohomology#GodementResolutionExists

  • 007Y: the presheaf F#F^# is a sheaf

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf

  • 0080: for a presheaf of sets FF , any map F→HF → H into a sheaf factors uniquely through F→F#F → F^#

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafifyUniversal

Derived result for this query

  • 0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf

    f: fsharpx: x
Other derived facts4

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: fx: x

  • FF is a sheaf of sets on XX

    f: fsharpx: x

  • 0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)

    f: fsharp

  • 0080: any map F→HF → H into a sheaf of sets factors uniquely as F→F#→HF → F^# → H

    f: fg: fsharp
006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms
fx
urn:case:stacks:sh:fx
FF is a sheaf of sets on XX
fx
urn:case:stacks:sh:fsharpx
0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf
fx
urn:case:stacks:sh:fsharpx
0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)
f
urn:case:stacks:sh:fsharp
0080: any map F→HF → H into a sheaf of sets factors uniquely as F→F#→HF → F^# → H
fg
urn:case:stacks:sh:furn:case:stacks:sh:fsharp

0 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion3 rules

  1. 1

    006T: a presheaf whose compatible families of sections do not glue is not a sheaf

    What is missing

    • 006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_ifsharpNot establishedthis is the missing one
    • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_Vfsharp, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing
    rule
  2. 2

    006T: a presheaf in which a glued section is not unique is not a sheaf

    What is missing

    • 006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open coveringfsharpNot establishedthis is the missing one
    • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_Vfsharp, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByFailedUniqueness
    rule
  3. 3

    006T: a presheaf refuted by an open covering is not a sheaf of sets on XX

    What is missing

    • FF is not a sheaf on XX : the sheaf condition 006T fails on some open coveringfsharp, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering
    rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 4 layer
query_evaluationabelian_sheaf_onrule_applicationSheafifyAbelianPresheafrule_applicationabelian_presheaf_on/sufficientassertionplus_constructionassertionpresheaf_of_sets_onassertionabelian_group_structure

Proof nodes: 10 · assertion 3, rule_application 5, constraint_check 1, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
presheaf_of_sets_on
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_group_structure
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_presheaf_on/sufficient:7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae
attributes
definition
concept
abelian_presheaf_on
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_presheaf_on
evidence
—
Identifier
7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
abelian_presheaf_on/sufficient
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
plus_construction
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:SheafificationIsSheaf:eaf6a9aa2d0b4a500dc473cfcc0200ff61ea9beb9b8b4c6f79c6657450c8afd0
attributes
—
conclusion
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
sheaf_of_sets_on
evidence
—
Identifier
eaf6a9aa2d0b4a500dc473cfcc0200ff61ea9beb9b8b4c6f79c6657450c8afd0
Type
rule_application
Premises
  • fact-1
  • fact-3
Rule
SheafificationIsSheaf
sourceAnchors
—
substitution
v0
Identifier
fsharp
Type
entity_ref
v1
Identifier
f
Type
entity_ref
v2
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a
attributes
—
conclusion
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_sheaf_on
evidence
—
Identifier
71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a
Type
rule_application
Premises
  • 7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae
  • fact-3
Rule
SheafifyAbelianPresheaf
sourceAnchors
—
substitution
v0
Identifier
fsharp
Type
entity_ref
v1
Identifier
f
Type
entity_ref
v2
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:GodementResolutionExists:c3529a34c74fd31d4c672c809a4ecb778e70d8af01b9f392fdbc88127dbc32ff
attributes
—
conclusion
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_flasque_resolution
evidence
—
Identifier
c3529a34c74fd31d4c672c809a4ecb778e70d8af01b9f392fdbc88127dbc32ff
Type
rule_application
Premises
  • 71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a
Rule
GodementResolutionExists
sourceAnchors
—
substitution
v0
Identifier
fsharp
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:SheafifyUniversal:08e13163bd01f4b2580ae4fbd1215e62a1d2eacd896fef56d26ee23b484b96a7
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    fsharp
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
universal_among_maps_to_sheaves
evidence
—
Identifier
08e13163bd01f4b2580ae4fbd1215e62a1d2eacd896fef56d26ee23b484b96a7
Type
rule_application
Premises
  • fact-1
  • fact-3
Rule
SheafifyUniversal
sourceAnchors
—
substitution
v0
Identifier
fsharp
Type
entity_ref
v1
Identifier
f
Type
entity_ref
v2
Identifier
x
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4
attributes
—
conclusion
constraint
abelian_presheaf_on/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4
Type
constraint_check
Premises
  • 7889dfde746f9ab8807eab1f0add64a003fcaf203a97fcbecbe717dcc9c62dae
  • fact-1
  • fact-2
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_sheaf_on
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • 71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "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": [],
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            "predicate": "urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"
          },
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          "sourceAnchors": [],
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            "v0": {
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          },
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          "id": "urn:proof:query:mcp",
          "kind": "query_evaluation",
          "premises": [
            "urn:proof:apply:SheafifyAbelianPresheaf:71a0ffc581c9d38c8ed87215ff07264a41e544119fcaff883a64071bf550b44a"
          ],
          "sourceAnchors": []
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      "roots": [
        "urn:proof:constraint:abelian_presheaf_on/necessary:d9e2399cdfdd101b1998dce82c1b5e113c483740c26e7506b19c323888ffd3c4",
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  "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"
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  "vulnerableTo": [
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      "label": "006T: a presheaf in which a glued section is not unique is not a sheaf",
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        "presheaf_of_sets_on(urn:case:stacks:sh:fsharp, urn:case:stacks:sh:x)"
      ],
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          "status": "NEITHER"
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      "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)"
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          "status": "NEITHER"
        }
      ],
      "rule": "NotSheafByRefutingCovering"
    }
  ]
}
SourcesExcerpts: 7

tag/006K

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A presheaf of abelian groups on XX or an abelian presheaf over XX is a presheaf of sets ℱ\mathcal{F} such that for each open U⊂XU \subset X the set ℱ(U)\mathcal{F}(U) is endowed with the structure of an abelian group, and such that all restriction maps ρVU\rho^U_V are homomorphisms of abelian groups, see Lemma above.

  • A morphism of abelian presheaves over XX φ:ℱ→𝒢\varphi : \mathcal{F} \to \mathcal{G} is a morphism of presheaves of sets which induces a homomorphism of abelian groups ℱ(U)→𝒢(U)\mathcal{F}(U) \to \mathcal{G}(U) for every open U⊂XU \subset X .

  • The category of presheaves of abelian groups on XX is denoted PAb(X)\textit{PAb}(X) .

Original data · JSON
JSONRead only
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  "contentHash": "sha256:fa7c3dec36920848465ed2902237f8efade0877767eebcee37676bb1bb8717d9",
  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER",
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  "locator": "tag/006K",
  "package": "urn:stacks:clir:sheaf-cohomology",
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    {
      "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}"
    }
  ]
}

tag/006T

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A sheaf ℱ\mathcal{F} of sets on XX is a presheaf of sets which satisfies the following additional property: Given any open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i and any collection of sections si∈ℱ(Ui)s_i \in \mathcal{F}(U_i) , i∈Ii \in I such that ∀i,j∈I\forall i, j\in I

    si|Ui∩Uj=sj|Ui∩Ujs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}

    there exists a unique section s∈ℱ(U)s \in \mathcal{F}(U) such that si=s|Uis_i = s|_{U_i} for all i∈Ii \in I .

  • A morphism of sheaves of sets is simply a morphism of presheaves of sets.

  • The category of sheaves of sets on XX is denoted Sh(X)\Sh(X) .

Original data · JSON
JSONRead only
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  "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}"
    }
  ]
}

tag/007Y

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

The presheaf ℱ#\mathcal{F}^{\#} is a sheaf.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0080

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let ℱ\mathcal{F} be a presheaf of sets on XX . Any map ℱ→𝒢\mathcal{F} \to \mathcal{G} into a sheaf of sets factors uniquely as ℱ→ℱ#→𝒢\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G} .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0085

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space. Let ℱ\mathcal{F} be an abelian presheaf on XX . Then there exists a unique structure of abelian sheaf on ℱ#\mathcal{F}^\# such that ℱ→ℱ#\mathcal{F} \to \mathcal{F}^\# is a morphism of abelian presheaves. Moreover, the following adjointness property holds

MorPAb(X)(ℱ,i(𝒢))=MorAb(X)(ℱ#,𝒢).\Mor_{\textit{PAb}(X)}(\mathcal{F}, i(\mathcal{G})) = \Mor_{\textit{Ab}(X)}(\mathcal{F}^\#, \mathcal{G}).
Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/01AD

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Introduction

In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of 𝐙―\underline{\mathbf{Z}} -modules. Basic references are , and . We work out what happens for sheaves of modules on ringed topoi in another chapter (see Modules on Sites, Section ), although there we will mostly just duplicate the discussion from this chapter.

Original data · JSON
JSONRead only
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  "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."
    }
  ]
}

tag/0FKS

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. For every sheaf of 𝒪X\mathcal{O}_X -modules ℱ\mathcal{F} there is a resolution

0→ℱ→f*f*ℱ→f*f*f*f*ℱ→f*f*f*f*f*f*ℱ→…0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots

functorial in ℱ\mathcal{F} such that each term f*f*…f*f*ℱf_*f^* \ldots f_*f^*\mathcal{F} is a flasque 𝒪X\mathcal{O}_X -module and such that for all x∈Xx \in X the map

ℱx[0]→((f*f*ℱ)x→(f*f*f*f*ℱ)x→(f*f*f*f*f*f*ℱ)x→…)\mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big)

is a homotopy equivalence in the category of complexes of 𝒪X,x\mathcal{O}_{X, x} -modules.

Original data · JSON
JSONRead only
{
  "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",
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    {
      "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}"
    }
  ]
}

Packages in the snapshot

  • Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
abelian_sheaf_on: TRUE_ONLY — установлено Выведено 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) Применены правила: GodementResolutionExists, SheafificationIsSheaf, SheafifyAbelianPresheaf, SheafifyUniversal, abelian_presheaf_on/sufficient Ответ 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) Ответ 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) Ответ 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) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (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…) proof-граф: 10 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

JSONRead only
{
  "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": [
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          ],
          "rule": "urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient",
          "sourceAnchors": [],
          "substitution": {
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              "id": "urn:case:stacks:sh:f",
              "kind": "entity_ref"
            },
            "v1": {
              "id": "urn:case:stacks:sh:x",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
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          "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",
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            },
            "v1": {
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              "kind": "entity_ref"
            }
          }
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        {
          "attributes": {},
          "conclusion": {
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              {
                "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": [
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          "sourceAnchors": [],
          "substitution": {
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              "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": [
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            "urn:proof:assert:urn:mcp:case#fact-2"
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          "sourceAnchors": [],
          "substitution": {
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              "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 · JSON

JSONRead only
{
  "evaluationBytes": 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Display metadata

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓

JSON · calculations, sources and exact data

JSONRead only
{
  "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",
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  "jurisdiction": "вне юрисдикции государства",
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}
evaluation SHA-256
sha256:660311d3d2095dac745534a21a0f23d25eee3b82a4c3839e12ec3e0ae349d7ad
Original data · JSON
JSONRead only
{
  "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
}

Из предпучка строится пучок: конструкция сохраняет структуру абелевой группы.

Condition

Универсальность пучкования

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

0080: any map F→HF → H into a sheaf of sets factors uniquely as F→F#→HF → F^# → H

ffsharp

Input facts

  • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: fx: x

  • G=F#G = F^# is the sheafification of the presheaf FF : sections over UU are compatible families of germs (Sheaves, Section 007X), with the canonical map F→F#F → F^#

    g: fsharpf: f

Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "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
}
Why this resultApplied rules and conditions

Derivation path4 steps

  1. 1

    FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    case fact
  2. 2

    G=F#G = F^# is the sheafification of the presheaf FF : sections over UU are compatible families of germs (Sheaves, Section 007X), with the canonical map F→F#F → F^#

    g: urn:case:stacks:sh:fsharp; f: urn:case:stacks:sh:f

    case fact
  3. 3

    0080: for a presheaf of sets FF , any map F→HF → H into a sheaf factors uniquely through F→F#F → F^#

    0080: any map F→HF → H into a sheaf of sets factors uniquely as F→F#→HF → F^# → H: f: urn:case:stacks:sh:f; g: urn:case:stacks:sh:fsharp

    tag 0080

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafifyUniversal
    rule
  4. 4

    Query evaluation

    query

verified by the engine: 2 · case fact: 2 · Full graph: 5 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0080: for a presheaf of sets FF , any map F→HF → H into a sheaf factors uniquely through F→F#F → F^#

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafifyUniversal
Other rules in the evaluation1

Applied in the overall evaluation, but not on the proof path for this answer.

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 007Y: the presheaf F#F^# is a sheaf

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf

Derived result for this query

  • 0080: any map F→HF → H into a sheaf of sets factors uniquely as F→F#→HF → F^# → H

    f: fg: fsharp
Other derived facts1

  • FF is a sheaf of sets on XX

    f: fsharpx: x
FF is a sheaf of sets on XX
fx
urn:case:stacks:sh:fsharpurn:case:stacks:sh:x
0080: any map F→HF → H into a sheaf of sets factors uniquely as F→F#→HF → F^# → H
fg
urn:case:stacks:sh:furn:case:stacks:sh:fsharp

0 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion3 rules

  1. 1

    006T: a presheaf whose compatible families of sections do not glue is not a sheaf

    What is missing

    • 006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_ifsharpNot establishedthis is the missing one
    • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_Vfsharp, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing
    rule
  2. 2

    006T: a presheaf in which a glued section is not unique is not a sheaf

    What is missing

    • 006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open coveringfsharpNot establishedthis is the missing one
    • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_Vfsharp, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByFailedUniqueness
    rule
  3. 3

    006T: a presheaf refuted by an open covering is not a sheaf of sets on XX

    What is missing

    • FF is not a sheaf on XX : the sheaf condition 006T fails on some open coveringfsharp, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering
    rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 3 layer
query_evaluationuniversal_among_maps_to_sheavesrule_applicationSheafifyUniversalassertionpresheaf_of_sets_onassertionplus_construction

Proof nodes: 5 · assertion 2, rule_application 2, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
presheaf_of_sets_on
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
plus_construction
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:SheafificationIsSheaf:af06faaad567ecfebddfcbec65edca5aaad2d911c935bbe8196ed80f8a0b22d2
attributes
—
conclusion
Arguments
  • Identifier
    fsharp
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
sheaf_of_sets_on
evidence
—
Identifier
af06faaad567ecfebddfcbec65edca5aaad2d911c935bbe8196ed80f8a0b22d2
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
SheafificationIsSheaf
sourceAnchors
—
substitution
v0
Identifier
fsharp
Type
entity_ref
v1
Identifier
f
Type
entity_ref
v2
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:SheafifyUniversal:7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    fsharp
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
universal_among_maps_to_sheaves
evidence
—
Identifier
7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
SheafifyUniversal
sourceAnchors
—
substitution
v0
Identifier
fsharp
Type
entity_ref
v1
Identifier
f
Type
entity_ref
v2
Identifier
x
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    fsharp
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
universal_among_maps_to_sheaves
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • 7d4d94c6095ac3eacf4b6206ef85c2263db401ffcb569905a6f09c98cbba0397
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "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"
  },
  "proofRef": "urn:proof:query:mcp",
  "rulesApplied": [
    "urn:stacks:clir:sheaf-cohomology#SheafificationIsSheaf",
    "urn:stacks:clir:sheaf-cohomology#SheafifyUniversal"
  ],
  "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"
    }
  ]
}
SourcesExcerpts: 3

tag/006T

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A sheaf ℱ\mathcal{F} of sets on XX is a presheaf of sets which satisfies the following additional property: Given any open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i and any collection of sections si∈ℱ(Ui)s_i \in \mathcal{F}(U_i) , i∈Ii \in I such that ∀i,j∈I\forall i, j\in I

    si|Ui∩Uj=sj|Ui∩Ujs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}

    there exists a unique section s∈ℱ(U)s \in \mathcal{F}(U) such that si=s|Uis_i = s|_{U_i} for all i∈Ii \in I .

  • A morphism of sheaves of sets is simply a morphism of presheaves of sets.

  • The category of sheaves of sets on XX is denoted Sh(X)\Sh(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/007Y

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

The presheaf ℱ#\mathcal{F}^{\#} is a sheaf.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0080

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let ℱ\mathcal{F} be a presheaf of sets on XX . Any map ℱ→𝒢\mathcal{F} \to \mathcal{G} into a sheaf of sets factors uniquely as ℱ→ℱ#→𝒢\mathcal{F} \to \mathcal{F}^\# \to \mathcal{G} .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

Packages in the snapshot

  • Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
universal_among_maps_to_sheaves: TRUE_ONLY — установлено Выведено 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) Применены правила: SheafificationIsSheaf, SheafifyUniversal Ответ 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) Ответ 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) Ответ 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) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (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…) proof-граф: 5 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

JSONRead only
{
  "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": [],
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        "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"
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      "rule": "NotSheafByRefutingCovering"
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Execution · JSON

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Display metadata

JSONRead only
{
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            "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)"
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        ],
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        "ruleId": "urn:stacks:clir:sheaf-cohomology#NotSheafByFailedGluing"
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      "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": []
}

JSON · calculations, sources and exact data

JSONRead only
{
  "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"
}
evaluation SHA-256
sha256:bb6b9c8bb5830382dc347c3d5d63d7c38df4cf39775a6d713774db2edbfbb2e9
Original data · JSON
JSONRead only
{
  "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
}

Пучкование универсально среди отображений предпучка в пучки: через него пропускается любое такое отображение.

Condition

H⁰ равна глобальным сечениям

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

01DZ: Hn(X,F)H^n(X, F) , the n-th cohomology group of the abelian sheaf FF , is the given object

x0fh0

Input facts

  • the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: abxx: x

  • the category is AbAb , the category of abelian groups

    ab: ab

  • the functor is Γ(X,−):Ab(X)→Ab,F↦Γ(X,F)=F(X)Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)

    g: gammax: x

  • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: fx: x

  • each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: f

  • 006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: f

  • 006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: f

  • Γ(X,F)=F(X)Γ(X, F) = F(X) is the given abelian group

    f: fx: xh: h0

Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "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
}
Why this resultApplied rules and conditions

Derivation path30 steps

  1. 1

    the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: urn:case:stacks:sh:abx; x: urn:case:stacks:sh:x

    case fact
  2. 2

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular additive

    0104: the category is additive: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXAdditive
    rule
  3. 3

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for every morphism

    the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCoimageImage
    rule
  4. 4

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all cokernels exist

    all cokernels exist in the category: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCokernels
    rule
  5. 5

    01DG: the category of abelian sheaves on a topological space XX has enough injectives

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ: c: urn:case:stacks:sh:abx

    tag 01DG

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives
    rule
  6. 6

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all kernels exist

    all kernels exist in the category: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXKernels
    rule
  7. 7

    0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    c: urn:case:stacks:sh:abx

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  8. 8

    the category is AbAb , the category of abelian groups

    ab: urn:case:stacks:sh:ab

    case fact
  9. 9

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular additive

    0104: the category is additive: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbAdditive
    rule
  10. 10

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism

    the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCoimageImage
    rule
  11. 11

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all cokernels exist

    all cokernels exist in the category: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCokernels
    rule
  12. 12

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all kernels exist

    all kernels exist in the category: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbKernels
    rule
  13. 13

    0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    c: urn:case:stacks:sh:ab

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  14. 14

    the functor is Γ(X,−):Ab(X)→Ab,F↦Γ(X,F)=F(X)Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)

    g: urn:case:stacks:sh:gamma; x: urn:case:stacks:sh:x

    case fact
  15. 15

    0716: Γ(X,−)Γ(X, −) is a functor from Ab(X)Ab(X) to AbAb

    the functor goes from the first category to the second: F:A→BF : A → B: f: urn:case:stacks:sh:gamma; a: urn:case:stacks:sh:abx; b: urn:case:stacks:sh:ab

    tag 0716

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween
    rule
  16. 16

    0716: Γ(X,−)Γ(X, −) is a left exact functor

    for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact: f: urn:case:stacks:sh:gamma

    tag 0716

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact
    rule
  17. 17

    010N (2): FF is left exact if for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact

    0034 (1): the functor is left exact: f: urn:case:stacks:sh:gamma

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactBySES
    rule
  18. 18

    010N (1): if FF is left exact, then it is additive

    the functor is additive: f: urn:case:stacks:sh:gamma

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive
    rule
  19. 19

    05TI (2): if AA is abelian with enough injectives and F:A→BF : A → B is an additive functor into an abelian category, then RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined: f: urn:case:stacks:sh:gamma

    tag 05TI, tag 05SV, tag 05T4

    Identifier
    urn:stacks:clir:derived-functors#RFEverywhereDefined
    rule
  20. 20

    05TD (3): if RFRF is everywhere defined and FF is left exact, then F→R0FF → R^0F is an isomorphism

    05TD (3): the map F→R0FF → R^0F is an isomorphism: f: urn:case:stacks:sh:gamma

    tag 05TD

    Identifier
    urn:stacks:clir:derived-functors#R0AgreesIfLeftExact
    rule
  21. 21

    FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    case fact
  22. 22

    each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: urn:case:stacks:sh:f

    case fact
  23. 23

    006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006K

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
    rule
  24. 24

    006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: urn:case:stacks:sh:f

    case fact
  25. 25

    006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: urn:case:stacks:sh:f

    case fact
  26. 26

    006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    FF is a sheaf of sets on XX: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing
    rule
  27. 27

    0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 0070

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition
    rule
  28. 28

    Γ(X,F)=F(X)Γ(X, F) = F(X) is the given abelian group

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x; h: urn:case:stacks:sh:h0

    case fact
  29. 29

    006E with 05TD (3): H0(X,F)=Γ(X,F)H^0(X, F) = Γ(X, F) since Γ(X,−)Γ(X, −) is left exact

    01DZ: Hn(X,F)H^n(X, F) , the n-th cohomology group of the abelian sheaf FF , is the given object: x: urn:case:stacks:sh:x; n: 0; f: urn:case:stacks:sh:f; h: urn:case:stacks:sh:h0

    tag 006E, tag 0716

    Identifier
    urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections
    rule
  30. 30

    Query evaluation

    query

verified by the engine: 22 · case fact: 8 · Full graph: 41 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

  • 010N (2): FF is left exact if for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact

    Identifier
    urn:stacks:clir:categories#LeftExactBySES

  • 010N (1): if FF is left exact, then it is additive

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive

  • 0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

  • 05TD (3): if RFRF is everywhere defined and FF is left exact, then F→R0FF → R^0F is an isomorphism

    Identifier
    urn:stacks:clir:derived-functors#R0AgreesIfLeftExact

  • 05TI (2): if AA is abelian with enough injectives and F:A→BF : A → B is an additive functor into an abelian category, then RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    Identifier
    urn:stacks:clir:derived-functors#RFEverywhereDefined
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbAdditive

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCoimageImage

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCokernels

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbKernels

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXAdditive

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for every morphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCoimageImage

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCokernels

  • 01DG: the category of abelian sheaves on a topological space XX has enough injectives

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXKernels

  • 0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition

  • 0716: Γ(X,−)Γ(X, −) is a functor from Ab(X)Ab(X) to AbAb

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween

  • 0716: Γ(X,−)Γ(X, −) is a left exact functor

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact

  • 006E with 05TD (3): H0(X,F)=Γ(X,F)H^0(X, F) = Γ(X, F) since Γ(X,−)Γ(X, −) is left exact

    Identifier
    urn:stacks:clir:sheaf-cohomology#H0IsGlobalSections

  • 006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
Other rules in the evaluation7

Applied in the overall evaluation, but not on the proof path for this answer.

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

  • 05TE (1): if RFRF is everywhere defined, the RiFR^iF come equipped with a canonical structure of a δδ -functor

    Identifier
    urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor

  • 015B (4): if AA is abelian with enough injectives and F:A→BF : A → B is left exact, then (RiF,δ)(R^iF, δ) is a universal δδ -functor

    Identifier
    urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor

  • 013K (1): in an abelian category with enough injectives any object has an injective resolution

    Identifier
    urn:stacks:clir:derived-functors#InjectiveResolutionsExist

  • 05TD (1): if RFRF is everywhere defined, then RiF=0R^iF = 0 for i<0i < 0

    Identifier
    urn:stacks:clir:derived-functors#NegativeDerivedVanish
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0070 (2): an abelian sheaf on XX is an object of Ab(X)Ab(X)

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX

  • 01DZ: the functors Hi(X,−)H^i(X, −) form a universal δδ -functor Ab(X)→AbAb(X) → Ab

    Identifier
    urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor

  • 0FKS with 01AD: every abelian sheaf FF on XX has the Godement resolution 0→F→f*f*F→…0 → F → f_*f^*F → … by flasque sheaves

    Identifier
    urn:stacks:clir:sheaf-cohomology#GodementResolutionExists

Derived result for this query

  • 01DZ: Hn(X,F)H^n(X, F) , the n-th cohomology group of the abelian sheaf FF , is the given object

    x: xn: 0f: fh: h0
Other derived facts7

  • 01DZ: the family Hi(X,−)H^i(X, −) forms a universal δδ -functor from Ab(X)Ab(X) to AbAb

    x: x

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: fx: x

  • FF is a sheaf of sets on XX

    f: fx: x

  • 0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf

    f: fx: x

  • the object belongs to the category: x∈Ob(C)x ∈ Ob(C)

    x: fc: abx

  • 0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)

    f: f

  • the object has an injective resolution

    a: f
01DZ: the family Hi(X,−)H^i(X, −) forms a universal δδ -functor from Ab(X)Ab(X) to AbAb
x
x
006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms
fx
urn:case:stacks:sh:fx
FF is a sheaf of sets on XX
fx
urn:case:stacks:sh:fx
0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf
fx
urn:case:stacks:sh:fx
the object belongs to the category: x∈Ob(C)x ∈ Ob(C)
xc
furn:case:stacks:sh:abx
0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)
f
urn:case:stacks:sh:f
the object has an injective resolution
a
urn:case:stacks:sh:f
01DZ: Hn(X,F)H^n(X, F) , the n-th cohomology group of the abelian sheaf FF , is the given object
xnfh
x0urn:case:stacks:sh:furn:case:stacks:sh:h0

20 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

  1. 1

    006T: a presheaf refuted by an open covering is not a sheaf of sets on XX

    What is missing

    • FF is not a sheaf on XX : the sheaf condition 006T fails on some open coveringf, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering
    rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 6 layer
query_evaluationsheaf_cohomologyrule_applicationH0IsGlobalSectionsrule_applicationAbelianSheafByDefinitionrule_applicationR0AgreesIfLeftExactassertionglobal_sections_functorassertionglobal_sectionsrule_applicationSheafByGluingrule_applicationabelian_presheaf_on/sufficientrule_applicationLeftExactBySESrule_applicationRFEverywhereDefinedassertionpresheaf_of_sets_onassertioncompatible_sections_glueassertiongluing_is_uniqueassertionabelian_group_structurerule_applicationGlobalSectionsLeftExactrule_applicationAbXEnoughInjectivesrule_applicationGlobalSectionsBetweenrule_applicationLeftExactIsAdditiverule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientassertionsheaves_categoryassertionabelian_groups_categoryrule_applicationAbAdditiverule_applicationAbCoimageImagerule_applicationAbCokernelsrule_applicationAbKernelsrule_applicationAbXAdditiverule_applicationAbXCoimageImagerule_applicationAbXCokernelsrule_applicationAbXKernels

Proof nodes: 41 · assertion 8, rule_application 28, constraint_check 4, query_evaluation 1

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Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON

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Download JSON ↓
SourcesExcerpts: 19

tag/006E

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A presheaf ℱ\mathcal{F} of sets on XX is a rule which assigns to each open U⊂XU \subset X a set ℱ(U)\mathcal{F}(U) and to each inclusion V⊂UV \subset U a map ρVU:ℱ(U)→ℱ(V)\rho^U_V : \mathcal{F}(U) \to \mathcal{F}(V) such that ρUU=idℱ(U)\rho^U_U = \text{id}_{\mathcal{F}(U)} and whenever W⊂V⊂UW \subset V \subset U we have ρWU=ρWV∘ρVU\rho^U_W = \rho^V_W \circ \rho ^U_V .

  • A morphism φ:ℱ→𝒢\varphi : \mathcal{F} \to \mathcal{G} of presheaves of sets on XX is a rule which assigns to each open U⊂XU \subset X a map of sets φ:ℱ(U)→𝒢(U)\varphi : \mathcal{F}(U) \to \mathcal{G}(U) compatible with restriction maps, i.e., whenever V⊂U⊂XV \subset U \subset X are open the diagram

    \xymatrix{
\mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho^U_V} &
\mathcal{G}(U) \ar[d]^{\rho^U_V} \\
\mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V)
}
    Диаграмма: исходный TeX

    commutes.

  • The category of presheaves of sets on XX will be denoted PSh(X)\textit{PSh}(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/006K

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A presheaf of abelian groups on XX or an abelian presheaf over XX is a presheaf of sets ℱ\mathcal{F} such that for each open U⊂XU \subset X the set ℱ(U)\mathcal{F}(U) is endowed with the structure of an abelian group, and such that all restriction maps ρVU\rho^U_V are homomorphisms of abelian groups, see Lemma above.

  • A morphism of abelian presheaves over XX φ:ℱ→𝒢\varphi : \mathcal{F} \to \mathcal{G} is a morphism of presheaves of sets which induces a homomorphism of abelian groups ℱ(U)→𝒢(U)\mathcal{F}(U) \to \mathcal{G}(U) for every open U⊂XU \subset X .

  • The category of presheaves of abelian groups on XX is denoted PAb(X)\textit{PAb}(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/006T

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A sheaf ℱ\mathcal{F} of sets on XX is a presheaf of sets which satisfies the following additional property: Given any open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i and any collection of sections si∈ℱ(Ui)s_i \in \mathcal{F}(U_i) , i∈Ii \in I such that ∀i,j∈I\forall i, j\in I

    si|Ui∩Uj=sj|Ui∩Ujs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}

    there exists a unique section s∈ℱ(U)s \in \mathcal{F}(U) such that si=s|Uis_i = s|_{U_i} for all i∈Ii \in I .

  • A morphism of sheaves of sets is simply a morphism of presheaves of sets.

  • The category of sheaves of sets on XX is denoted Sh(X)\Sh(X) .

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      "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}"
    }
  ]
}

tag/0070

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • An abelian sheaf on XX or sheaf of abelian groups on XX is an abelian presheaf on XX such that the underlying presheaf of sets is a sheaf.

  • The category of sheaves of abelian groups is denoted Ab(X)\textit{Ab}(X) .

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  "contentHash": "sha256:04f51a87121abeee45cd4e99f1379db299b359b4ac801017c7770a7f05d2c370",
  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_SHEAVES_MASTER",
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  "id": "urn:stacks:clir:sheaf-cohomology#ST_0070",
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      "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}"
    }
  ]
}

tag/01AD

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Introduction

In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of 𝐙―\underline{\mathbf{Z}} -modules. Basic references are , and . We work out what happens for sheaves of modules on ringed topoi in another chapter (see Modules on Sites, Section ), although there we will mostly just duplicate the discussion from this chapter.

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  "contentHash": "sha256:d0262bf7656850a1933eef167a360437aa2bb6436e57461a3d23de09e8cc8ef3",
  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER",
  "fragmentKind": "section",
  "id": "urn:stacks:clir:sheaf-cohomology#ST_01AD",
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      "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."
    }
  ]
}

tag/01AG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is an abelian category. Moreover a complex

ℱ→𝒢→ℋ\mathcal{F} \to \mathcal{G} \to \mathcal{H}

is exact at 𝒢\mathcal{G} if and only if for all x∈Xx \in X the complex

ℱx→𝒢x→ℋx\mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x

is exact at 𝒢x\mathcal{G}_x .

Original data · JSON
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      "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}"
    }
  ]
}

tag/01DG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space. The category of abelian sheaves on XX has enough injectives. In fact it has functorial injective embeddings.

Original data · JSON
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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_INJECTIVES_MASTER",
  "fragmentKind": "lemma",
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      "contentHash": "sha256:e4b4f09acb0a39efab590f7a0a1974dde000a27f6205d5284eb9b41b91b7f6d6",
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      "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}"
    }
  ]
}

tag/01DZ

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Cohomology of sheaves

Let XX be a topological space. Let ℱ\mathcal{F} be an abelian sheaf. We know that the category of abelian sheaves on XX has enough injectives, see Injectives, Lemma . Hence we can choose an injective resolution ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet . As is customary we define H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet)) to be the ii th cohomology group of the abelian sheaf ℱ\mathcal{F} . The family of functors Hi(X,−)H^i(X, -) forms a universal δ\delta -functor from Ab(X)→Ab\textit{Ab}(X) \to \textit{Ab} . Let f:X→Yf : X \to Y be a continuous map of topological spaces. With ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet as above we define R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) to be the ii th higher direct image of ℱ\mathcal{F} . The family of functors Rif*R^if_* forms a universal δ\delta -functor from Ab(X)→Ab(Y)\textit{Ab}(X) \to \textit{Ab}(Y) . Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. Let ℱ\mathcal{F} be an 𝒪X\mathcal{O}_X -module. We know that the category of 𝒪X\mathcal{O}_X -modules on XX has enough injectives, see Injectives, Lemma . Hence we can choose an injective resolution ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet . As is customary we define H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet)) to be the ii th cohomology group of ℱ\mathcal{F} . The family of functors Hi(X,−)H^i(X, -) forms a universal δ\delta -functor from Mod(𝒪X)→Mod𝒪X(X)\textit{Mod}(\mathcal{O}_X) \to \text{Mod}_{\mathcal{O}_X(X)} . Let f:(X,𝒪X)→(Y,𝒪Y)f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y) be a morphism of ringed spaces. With ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet as above we define R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) to be the ii th higher direct image of ℱ\mathcal{F} . The family of functors Rif*R^if_* forms a universal δ\delta -functor from Mod(𝒪X)→Mod(𝒪Y)\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y) .

Original data · JSON
JSONRead only
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  "contentHash": "sha256:b1dd6e78bfe8178dd35b60b16af0e740bb77f76f64096d01aaa44a811d4491b3",
  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
  "fragmentKind": "section",
  "id": "urn:stacks:clir:sheaf-cohomology#ST_01DZ",
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    {
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      "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 $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$."
    }
  ]
}

tag/0716

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Derived functors

We briefly explain how to get right derived functors using resolution functors. For the unbounded derived functors, please see Section . Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is abelian, see Modules, Lemma . In this chapter we will write

K(𝒪X)=K(Mod(𝒪X))andD(𝒪X)=D(Mod(𝒪X)).K(\mathcal{O}_X) = K(\textit{Mod}(\mathcal{O}_X)) \quad \text{and} \quad D(\mathcal{O}_X) = D(\textit{Mod}(\mathcal{O}_X)).

and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition and Definition . By Derived Categories, Remark there exists a resolution functor

j=jX:K+(Mod(𝒪X))⟶K+(ℐ)j = j_X : K^{+}(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K^{+}(\mathcal{I})

where ℐ\mathcal{I} is the strictly full additive subcategory of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) consisting of injective sheaves. For any left exact functor F:Mod(𝒪X)→ℬF : \textit{Mod}(\mathcal{O}_X) \to \mathcal{B} into any abelian category ℬ\mathcal{B} we will denote RFRF the right derived functor described in Derived Categories, Section and constructed using the resolution functor jXj_X just described: RF = F \circ j_X' : D^{+}(X) \longrightarrow D^{+}(\mathcal{B}) see Derived Categories, Lemma for notation. Note that we may think of RFRF as defined on Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) , Comp+(Mod(𝒪X))\text{Comp}^{+}(\textit{Mod}(\mathcal{O}_X)) , K+(X)K^{+}(X) , or D+(X)D^{+}(X) depending on the situation. According to Derived Categories, Definition we obtain the ii th right derived functor R^iF = H^i \circ RF : Mod (\mathcal{O}_X) \longrightarrow \mathcal{B} so that R0F=FR^0F = F and {RiF,δ}i≥0\{R^iF, \delta\}_{i \geq 0} is universal δ\delta -functor, see Derived Categories, Lemma . Here are two special cases of this construction. Given a ring RR we write K(R)=K(ModR)K(R) = K(\text{Mod}_R) and D(R)=D(ModR)D(R) = D(\text{Mod}_R) and similarly for bounded versions. For any open U⊂XU \subset X we have a left exact functor Γ(U,−):Mod(𝒪X)⟶Mod𝒪X(U)\Gamma(U, -) : \textit{Mod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)} which gives rise to R\Gamma(U, -) : D^{+}(X) \longrightarrow D^{+}(\mathcal{O}_X(U)) by the discussion above. We set Hi(U,−)=RiΓ(U,−)H^i(U, -) = R^i\Gamma(U, -) . If U=XU = X we recover (). If f:X→Yf : X \to Y is a morphism of ringed spaces, then we have the left exact functor f*:Mod(𝒪X)⟶Mod(𝒪Y)f_* : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_Y) which gives rise to the derived pushforward Rf_* : D^{+}(X) \longrightarrow D^{+}(Y) The ii th cohomology sheaf of Rf*ℱ•Rf_*\mathcal{F}^\bullet is denoted Rif*ℱ•R^if_*\mathcal{F}^\bullet and called the ii th higher direct image in accordance with (). The two displayed functors above are exact functors of derived categories. Abuse of notation: When the functor Rf*Rf_* , or any other derived functor, is applied to a sheaf ℱ\mathcal{F} on XX or a complex of sheaves it is understood that ℱ\mathcal{F} has been replaced by a suitable resolution of ℱ\mathcal{F} . To facilitate this kind of operation we will say, given an object ℱ•∈D(𝒪X)\mathcal{F}^\bullet \in D(\mathcal{O}_X) , that a bounded below complex ℐ•\mathcal{I}^\bullet of injectives of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) represents ℱ•\mathcal{F}^\bullet in the derived category if there exists a quasi-isomorphism ℱ•→ℐ•\mathcal{F}^\bullet \to \mathcal{I}^\bullet . In the same vein the phrase ``let α:ℱ•→𝒢•\alpha : \mathcal{F}^\bullet \to \mathcal{G}^\bullet be a morphism of D(𝒪X)D(\mathcal{O}_X) '' does not mean that α\alpha is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.

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      "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."
    }
  ]
}

tag/0FKS

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. For every sheaf of 𝒪X\mathcal{O}_X -modules ℱ\mathcal{F} there is a resolution

0→ℱ→f*f*ℱ→f*f*f*f*ℱ→f*f*f*f*f*f*ℱ→…0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots

functorial in ℱ\mathcal{F} such that each term f*f*…f*f*ℱf_*f^* \ldots f_*f^*\mathcal{F} is a flasque 𝒪X\mathcal{O}_X -module and such that for all x∈Xx \in X the map

ℱx[0]→((f*f*ℱ)x→(f*f*f*f*ℱ)x→(f*f*f*f*f*f*ℱ)x→…)\mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big)

is a homotopy equivalence in the category of complexes of 𝒪X,x\mathcal{O}_{X, x} -modules.

Original data · JSON
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  "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",
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    {
      "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}"
    }
  ]
}

tag/0109

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

A category 𝒜\mathcal{A} is abelian if it is additive, if all kernels and cokernels exist, and if the natural map Coim(f)→Im(f)\Coim(f) \to \Im(f) is an isomorphism for all morphisms ff of 𝒜\mathcal{A} .

Original data · JSON
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  "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",
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      "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"
}

tag/010N

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} and ℬ\mathcal{B} be abelian categories. Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be a functor.

  • If FF is either left or right exact, then it is additive.

  • FF is left exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence 0→F(A)→F(B)→F(C)0 \to F(A) \to F(B) \to F(C) is exact.

  • FF is right exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence F(A)→F(B)→F(C)→0F(A) \to F(B) \to F(C) \to 0 is exact.

  • FF is exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence 0→F(A)→F(B)→F(C)→00 \to F(A) \to F(B) \to F(C) \to 0 is exact.

Original data · JSON
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  "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",
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    {
      "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"
}

tag/013K

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category. Assume 𝒜\mathcal{A} has enough injectives.

  • Any object of 𝒜\mathcal{A} has an injective resolution.

  • If Hn(K•)=0H^n(K^\bullet) = 0 for all n≪0n \ll 0 then K•K^\bullet has an injective resolution.

  • If K•K^\bullet is a complex with Kn=0K^n = 0 for n<an < a , then there exists an injective resolution α:K•→I•\alpha : K^\bullet \to I^\bullet with In=0I^n = 0 for n<an < a such that each αn:Kn→In\alpha^n : K^n \to I^n is injective.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/015B

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category with enough injectives. Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be a left exact functor.

  • For any short exact sequence 0→A•→B•→C•→00 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 of complexes in Comp+(𝒜)\text{Comp}^{+}(\mathcal{A}) there is an associated long exact sequence

    …→Hi(RF(A•))→Hi(RF(B•))→Hi(RF(C•))→Hi+1(RF(A•))→…\ldots \to H^i(RF(A^\bullet)) \to H^i(RF(B^\bullet)) \to H^i(RF(C^\bullet)) \to H^{i + 1}(RF(A^\bullet)) \to \ldots

  • The functors RiF:𝒜→ℬR^iF : \mathcal{A} \to \mathcal{B} are zero for i<0i < 0 . Also R0F=F:𝒜→ℬR^0F = F : \mathcal{A} \to \mathcal{B} .

  • We have RiF(I)=0R^iF(I) = 0 for i>0i > 0 and II injective.

  • The sequence (RiF,δ)(R^iF, \delta) forms a universal δ\delta -functor (see Homology, Definition ) from 𝒜\mathcal{A} to ℬ\mathcal{B} .

Original data · JSON
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{
  "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}"
    }
  ]
}

tag/05SV

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

In Situation . We say FF is right derivable , or that RFRF everywhere defined if RFRF is defined at every object of 𝒟\mathcal{D} . We say FF is left derivable , or that LFLF everywhere defined if LFLF is defined at every object of 𝒟\mathcal{D} .

Original data · JSON
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  "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",
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    {
      "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}"
    }
  ]
}

tag/05T4

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Here F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} is an additive functor between abelian categories. This induces exact functors

F:K(𝒜)→K(ℬ),K+(𝒜)→K+(ℬ),K−(𝒜)→K−(ℬ).F : K(\mathcal{A}) \to K(\mathcal{B}), \quad K^{+}(\mathcal{A}) \to K^{+}(\mathcal{B}), \quad K^{-}(\mathcal{A}) \to K^{-}(\mathcal{B}).

See Lemma . We also denote FF the composition K(𝒜)→D(ℬ)K(\mathcal{A}) \to D(\mathcal{B}) , K+(𝒜)→D+(ℬ)K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) , and K−(𝒜)→D−(ℬ)K^{-}(\mathcal{A}) \to D^-(\mathcal{B}) of FF with the localization functor K(ℬ)→D(ℬ)K(\mathcal{B}) \to D(\mathcal{B}) , etc. This situation leads to four derived functors we will consider in the following.

  • The right derived functor of F:K(𝒜)→D(ℬ)F : K(\mathcal{A}) \to D(\mathcal{B}) relative to the multiplicative system Qis(𝒜)\text{Qis}(\mathcal{A}) .

  • The right derived functor of F:K+(𝒜)→D+(ℬ)F : K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) relative to the multiplicative system Qis+(𝒜)\text{Qis}^{+}(\mathcal{A}) .

  • The left derived functor of F:K(𝒜)→D(ℬ)F : K(\mathcal{A}) \to D(\mathcal{B}) relative to the multiplicative system Qis(𝒜)\text{Qis}(\mathcal{A}) .

  • The left derived functor of F:K−(𝒜)→D−(ℬ)F : K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B}) relative to the multiplicative system Qis−(𝒜)\text{Qis}^-(\mathcal{A}) . Each of these cases is an example of Situation .

Original data · JSON
JSONRead only
{
  "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",
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    {
      "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}"
    }
  ]
}

tag/05TD

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be an additive functor between abelian categories and assume RF:D+(𝒜)→D+(ℬ)RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) is everywhere defined.

  • We have RiF=0R^iF = 0 for i<0i < 0 ,

  • R0FR^0F is left exact,

  • the map F→R0FF \to R^0F is an isomorphism if and only if FF is left exact.

Original data · JSON
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  "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}"
    }
  ]
}

tag/05TE

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be an additive functor between abelian categories and assume RF:D+(𝒜)→D+(ℬ)RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) is everywhere defined.

  • The functors RiFR^iF , i≥0i \geq 0 come equipped with a canonical structure of a δ\delta -functor from 𝒜→ℬ\mathcal{A} \to \mathcal{B} , see Homology, Definition .

  • If every object of 𝒜\mathcal{A} is a subobject of a right acyclic object for FF , then {RiF,δ}i≥0\{R^iF, \delta\}_{i \geq 0} is a universal δ\delta -functor, see Homology, Definition .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/05TI

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category with enough injectives.

  • For any exact functor F:K+(𝒜)→𝒟F : K^{+}(\mathcal{A}) \to \mathcal{D} into a triangulated category 𝒟\mathcal{D} the right derived functor

    RF:D+(𝒜)⟶𝒟RF : D^{+}(\mathcal{A}) \longrightarrow \mathcal{D}

    is everywhere defined.

  • For any additive functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} into an abelian category ℬ\mathcal{B} the right derived functor

    RF:D+(𝒜)⟶D+(ℬ)RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B})

    is everywhere defined.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

Packages in the snapshot

  • Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
  • 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
sheaf_cohomology: TRUE_ONLY — установлено Выведено 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) …и ещё 20 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены 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 Ответ 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) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (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…) Вместе 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 proof-граф: 41 узлов — поле evaluation готово для law_explain

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{
  "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"
}
evaluation SHA-256
sha256:77517a1cb5de03a401c900ab80ad61a47a5b28fff69456e978b42f0a27aafb4d
Original data · JSON
JSONRead only
{
  "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
}

Функтор глобальных сечений точен слева, поэтому нулевая степень его производного равна ему самому.

Condition

Старшие когомологии вялого пучка нулевые

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

Hi(X,F)=0H^i(X, F) = 0 for all i>0i > 0

xf

Input facts

  • the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: abxx: x

  • the category is AbAb , the category of abelian groups

    ab: ab

  • the functor is Γ(X,−):Ab(X)→Ab,F↦Γ(X,F)=F(X)Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)

    g: gammax: x

  • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: fx: x

  • each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: f

  • 006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: f

  • 006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: f

  • 09SW: for every U⊂VU ⊂ V open in XX the restriction map F(V)→F(U)F(V) → F(U) is surjective

    f: f

Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "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
}
Why this resultApplied rules and conditions

Derivation path32 steps

  1. 1

    the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: urn:case:stacks:sh:abx; x: urn:case:stacks:sh:x

    case fact
  2. 2

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular additive

    0104: the category is additive: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXAdditive
    rule
  3. 3

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for every morphism

    the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCoimageImage
    rule
  4. 4

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all cokernels exist

    all cokernels exist in the category: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCokernels
    rule
  5. 5

    01DG: the category of abelian sheaves on a topological space XX has enough injectives

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ: c: urn:case:stacks:sh:abx

    tag 01DG

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives
    rule
  6. 6

    01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all kernels exist

    all kernels exist in the category: c: urn:case:stacks:sh:abx

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXKernels
    rule
  7. 7

    0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    c: urn:case:stacks:sh:abx

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  8. 8

    the category is AbAb , the category of abelian groups

    ab: urn:case:stacks:sh:ab

    case fact
  9. 9

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular additive

    0104: the category is additive: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbAdditive
    rule
  10. 10

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism

    the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCoimageImage
    rule
  11. 11

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all cokernels exist

    all cokernels exist in the category: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCokernels
    rule
  12. 12

    01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all kernels exist

    all kernels exist in the category: c: urn:case:stacks:sh:ab

    tag 01AG, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbKernels
    rule
  13. 13

    0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    c: urn:case:stacks:sh:ab

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  14. 14

    the functor is Γ(X,−):Ab(X)→Ab,F↦Γ(X,F)=F(X)Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)

    g: urn:case:stacks:sh:gamma; x: urn:case:stacks:sh:x

    case fact
  15. 15

    0716: Γ(X,−)Γ(X, −) is a functor from Ab(X)Ab(X) to AbAb

    the functor goes from the first category to the second: F:A→BF : A → B: f: urn:case:stacks:sh:gamma; a: urn:case:stacks:sh:abx; b: urn:case:stacks:sh:ab

    tag 0716

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween
    rule
  16. 16

    0716: Γ(X,−)Γ(X, −) is a left exact functor

    for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact: f: urn:case:stacks:sh:gamma

    tag 0716

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact
    rule
  17. 17

    010N (2): FF is left exact if for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact

    0034 (1): the functor is left exact: f: urn:case:stacks:sh:gamma

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactBySES
    rule
  18. 18

    010N (1): if FF is left exact, then it is additive

    the functor is additive: f: urn:case:stacks:sh:gamma

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive
    rule
  19. 19

    05TI (2): if AA is abelian with enough injectives and F:A→BF : A → B is an additive functor into an abelian category, then RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined: f: urn:case:stacks:sh:gamma

    tag 05TI, tag 05SV, tag 05T4

    Identifier
    urn:stacks:clir:derived-functors#RFEverywhereDefined
    rule
  20. 20

    FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    case fact
  21. 21

    each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: urn:case:stacks:sh:f

    case fact
  22. 22

    006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006K

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
    rule
  23. 23

    006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: urn:case:stacks:sh:f

    case fact
  24. 24

    006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: urn:case:stacks:sh:f

    case fact
  25. 25

    006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    FF is a sheaf of sets on XX: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing
    rule
  26. 26

    0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 0070

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition
    rule
  27. 27

    09SW: for every U⊂VU ⊂ V open in XX the restriction map F(V)→F(U)F(V) → F(U) is surjective

    f: urn:case:stacks:sh:f

    case fact
  28. 28

    09SW: a presheaf is flasque if all restriction maps F(V)→F(U)F(V) → F(U) , U⊂VU ⊂ V open, are surjective

    09SW: the sheaf is flasque (flabby): f: urn:case:stacks:sh:f

    tag 09SW

    Identifier
    urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions
    rule
  29. 29

    09SY with 01AD: a flasque abelian sheaf is right acyclic for Γ(X,−)Γ(X, −)

    0157 (3): the object AA is right acyclic for FF : A[0]A[0] computes RFRF: a: urn:case:stacks:sh:f; f: urn:case:stacks:sh:gamma

    tag 09SY, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic
    rule
  30. 30

    015C (2): if FF is left exact, RFRF is everywhere defined and AA is right acyclic for FF , then RiF(A)=0R^iF(A) = 0 for all i>0i > 0

    015C: RiF(A)=0R^iF(A) = 0 for all i>0i > 0: f: urn:case:stacks:sh:gamma; a: urn:case:stacks:sh:f

    tag 015C

    Identifier
    urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic
    rule
  31. 31

    09SY with 015C (2): if FF is right acyclic for the left exact Γ(X,−)Γ(X, −) , then Hi(X,F)=RiΓ(X,−)(F)=0H^i(X, F) = R^iΓ(X, −)(F) = 0 for i>0i > 0

    Hi(X,F)=0H^i(X, F) = 0 for all i>0i > 0: x: urn:case:stacks:sh:x; f: urn:case:stacks:sh:f

    tag 09SY, tag 01DZ

    Identifier
    urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic
    rule
  32. 32

    Query evaluation

    query

verified by the engine: 24 · case fact: 8 · Full graph: 44 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

  • 010N (2): FF is left exact if for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact

    Identifier
    urn:stacks:clir:categories#LeftExactBySES

  • 010N (1): if FF is left exact, then it is additive

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive

  • 0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

  • 015C (2): if FF is left exact, RFRF is everywhere defined and AA is right acyclic for FF , then RiF(A)=0R^iF(A) = 0 for all i>0i > 0

    Identifier
    urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic

  • 05TI (2): if AA is abelian with enough injectives and F:A→BF : A → B is an additive functor into an abelian category, then RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    Identifier
    urn:stacks:clir:derived-functors#RFEverywhereDefined
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbAdditive

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCoimageImage

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCokernels

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbKernels

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXAdditive

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for every morphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCoimageImage

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCokernels

  • 01DG: the category of abelian sheaves on a topological space XX has enough injectives

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXKernels

  • 0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition

  • 09SY with 015C (2): if FF is right acyclic for the left exact Γ(X,−)Γ(X, −) , then Hi(X,F)=RiΓ(X,−)(F)=0H^i(X, F) = R^iΓ(X, −)(F) = 0 for i>0i > 0

    Identifier
    urn:stacks:clir:sheaf-cohomology#CohomologyVanishesForAcyclic

  • 09SW: a presheaf is flasque if all restriction maps F(V)→F(U)F(V) → F(U) , U⊂VU ⊂ V open, are surjective

    Identifier
    urn:stacks:clir:sheaf-cohomology#FlasqueByRestrictions

  • 09SY with 01AD: a flasque abelian sheaf is right acyclic for Γ(X,−)Γ(X, −)

    Identifier
    urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic

  • 0716: Γ(X,−)Γ(X, −) is a functor from Ab(X)Ab(X) to AbAb

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween

  • 0716: Γ(X,−)Γ(X, −) is a left exact functor

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact

  • 006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
Other rules in the evaluation8

Applied in the overall evaluation, but not on the proof path for this answer.

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

  • 05TE (1): if RFRF is everywhere defined, the RiFR^iF come equipped with a canonical structure of a δδ -functor

    Identifier
    urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor

  • 015B (4): if AA is abelian with enough injectives and F:A→BF : A → B is left exact, then (RiF,δ)(R^iF, δ) is a universal δδ -functor

    Identifier
    urn:stacks:clir:derived-functors#DerivedUniversalDeltaFunctor

  • 013K (1): in an abelian category with enough injectives any object has an injective resolution

    Identifier
    urn:stacks:clir:derived-functors#InjectiveResolutionsExist

  • 05TD (1): if RFRF is everywhere defined, then RiF=0R^iF = 0 for i<0i < 0

    Identifier
    urn:stacks:clir:derived-functors#NegativeDerivedVanish

  • 05TD (3): if RFRF is everywhere defined and FF is left exact, then F→R0FF → R^0F is an isomorphism

    Identifier
    urn:stacks:clir:derived-functors#R0AgreesIfLeftExact
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0070 (2): an abelian sheaf on XX is an object of Ab(X)Ab(X)

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX

  • 01DZ: the functors Hi(X,−)H^i(X, −) form a universal δδ -functor Ab(X)→AbAb(X) → Ab

    Identifier
    urn:stacks:clir:sheaf-cohomology#CohomologyUniversalDeltaFunctor

  • 0FKS with 01AD: every abelian sheaf FF on XX has the Godement resolution 0→F→f*f*F→…0 → F → f_*f^*F → … by flasque sheaves

    Identifier
    urn:stacks:clir:sheaf-cohomology#GodementResolutionExists

Derived result for this query

  • Hi(X,F)=0H^i(X, F) = 0 for all i>0i > 0

    x: xf: f
Other derived facts10

  • 01DZ: the family Hi(X,−)H^i(X, −) forms a universal δδ -functor from Ab(X)Ab(X) to AbAb

    x: x

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: fx: x

  • FF is a sheaf of sets on XX

    f: fx: x

  • 0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf

    f: fx: x

  • the object belongs to the category: x∈Ob(C)x ∈ Ob(C)

    x: fc: abx

  • 0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)

    f: f

  • the object has an injective resolution

    a: f

  • 09SW: the sheaf is flasque (flabby)

    f: f

  • 0157 (3): the object AA is right acyclic for FF : A[0]A[0] computes RFRF

    a: ff: gamma

  • 015C: RiF(A)=0R^iF(A) = 0 for all i>0i > 0

    f: gammaa: f
01DZ: the family Hi(X,−)H^i(X, −) forms a universal δδ -functor from Ab(X)Ab(X) to AbAb
x
x
006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms
fx
urn:case:stacks:sh:fx
FF is a sheaf of sets on XX
fx
urn:case:stacks:sh:fx
0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf
fx
urn:case:stacks:sh:fx
the object belongs to the category: x∈Ob(C)x ∈ Ob(C)
xc
furn:case:stacks:sh:abx
0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)
f
urn:case:stacks:sh:f
the object has an injective resolution
a
urn:case:stacks:sh:f
09SW: the sheaf is flasque (flabby)
f
urn:case:stacks:sh:f
0157 (3): the object AA is right acyclic for FF : A[0]A[0] computes RFRF
af
urn:case:stacks:sh:furn:case:stacks:sh:gamma
015C: RiF(A)=0R^iF(A) = 0 for all i>0i > 0
fa
urn:case:stacks:sh:gammaurn:case:stacks:sh:f
Hi(X,F)=0H^i(X, F) = 0 for all i>0i > 0
xf
xurn:case:stacks:sh:f

20 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

  1. 1

    006T: a presheaf refuted by an open covering is not a sheaf of sets on XX

    What is missing

    • FF is not a sheaf on XX : the sheaf condition 006T fails on some open coveringf, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering
    rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 7 layer
query_evaluationcohomology_vanishes_positiverule_applicationCohomologyVanishesForAcyclicrule_applicationHigherDerivedVanishForAcyclicassertionglobal_sections_functorrule_applicationFlasqueIsAcyclicrule_applicationLeftExactBySESrule_applicationRFEverywhereDefinedrule_applicationAbelianSheafByDefinitionrule_applicationFlasqueByRestrictionsrule_applicationGlobalSectionsLeftExactrule_applicationAbXEnoughInjectivesrule_applicationGlobalSectionsBetweenrule_applicationLeftExactIsAdditiverule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientrule_applicationSheafByGluingrule_applicationabelian_presheaf_on/sufficientassertionrestrictions_surjectiveassertionsheaves_categoryassertionabelian_groups_categoryrule_applicationAbAdditiverule_applicationAbCoimageImagerule_applicationAbCokernelsrule_applicationAbKernelsrule_applicationAbXAdditiverule_applicationAbXCoimageImagerule_applicationAbXCokernelsrule_applicationAbXKernelsassertionpresheaf_of_sets_onassertioncompatible_sections_glueassertiongluing_is_uniqueassertionabelian_group_structure

Proof nodes: 44 · assertion 8, rule_application 31, constraint_check 4, query_evaluation 1

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Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON

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SourcesExcerpts: 21

tag/006K

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A presheaf of abelian groups on XX or an abelian presheaf over XX is a presheaf of sets ℱ\mathcal{F} such that for each open U⊂XU \subset X the set ℱ(U)\mathcal{F}(U) is endowed with the structure of an abelian group, and such that all restriction maps ρVU\rho^U_V are homomorphisms of abelian groups, see Lemma above.

  • A morphism of abelian presheaves over XX φ:ℱ→𝒢\varphi : \mathcal{F} \to \mathcal{G} is a morphism of presheaves of sets which induces a homomorphism of abelian groups ℱ(U)→𝒢(U)\mathcal{F}(U) \to \mathcal{G}(U) for every open U⊂XU \subset X .

  • The category of presheaves of abelian groups on XX is denoted PAb(X)\textit{PAb}(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/006T

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A sheaf ℱ\mathcal{F} of sets on XX is a presheaf of sets which satisfies the following additional property: Given any open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i and any collection of sections si∈ℱ(Ui)s_i \in \mathcal{F}(U_i) , i∈Ii \in I such that ∀i,j∈I\forall i, j\in I

    si|Ui∩Uj=sj|Ui∩Ujs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}

    there exists a unique section s∈ℱ(U)s \in \mathcal{F}(U) such that si=s|Uis_i = s|_{U_i} for all i∈Ii \in I .

  • A morphism of sheaves of sets is simply a morphism of presheaves of sets.

  • The category of sheaves of sets on XX is denoted Sh(X)\Sh(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0070

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • An abelian sheaf on XX or sheaf of abelian groups on XX is an abelian presheaf on XX such that the underlying presheaf of sets is a sheaf.

  • The category of sheaves of abelian groups is denoted Ab(X)\textit{Ab}(X) .

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      "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}"
    }
  ]
}

tag/01AD

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Introduction

In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of 𝐙―\underline{\mathbf{Z}} -modules. Basic references are , and . We work out what happens for sheaves of modules on ringed topoi in another chapter (see Modules on Sites, Section ), although there we will mostly just duplicate the discussion from this chapter.

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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_MODULES_MASTER",
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  "id": "urn:stacks:clir:sheaf-cohomology#ST_01AD",
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      "contentHash": "sha256:3237316f9dec6bd87194ad43833111dbdf86cade5633082fbc0e7d50b3e2e7cf",
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      "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."
    }
  ]
}

tag/01AG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is an abelian category. Moreover a complex

ℱ→𝒢→ℋ\mathcal{F} \to \mathcal{G} \to \mathcal{H}

is exact at 𝒢\mathcal{G} if and only if for all x∈Xx \in X the complex

ℱx→𝒢x→ℋx\mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x

is exact at 𝒢x\mathcal{G}_x .

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  "id": "urn:stacks:clir:sheaf-cohomology#ST_01AG",
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      "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}"
    }
  ]
}

tag/01DG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space. The category of abelian sheaves on XX has enough injectives. In fact it has functorial injective embeddings.

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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_INJECTIVES_MASTER",
  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:sheaf-cohomology#ST_01DG",
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      "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}"
    }
  ]
}

tag/01DZ

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Cohomology of sheaves

Let XX be a topological space. Let ℱ\mathcal{F} be an abelian sheaf. We know that the category of abelian sheaves on XX has enough injectives, see Injectives, Lemma . Hence we can choose an injective resolution ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet . As is customary we define H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet)) to be the ii th cohomology group of the abelian sheaf ℱ\mathcal{F} . The family of functors Hi(X,−)H^i(X, -) forms a universal δ\delta -functor from Ab(X)→Ab\textit{Ab}(X) \to \textit{Ab} . Let f:X→Yf : X \to Y be a continuous map of topological spaces. With ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet as above we define R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) to be the ii th higher direct image of ℱ\mathcal{F} . The family of functors Rif*R^if_* forms a universal δ\delta -functor from Ab(X)→Ab(Y)\textit{Ab}(X) \to \textit{Ab}(Y) . Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. Let ℱ\mathcal{F} be an 𝒪X\mathcal{O}_X -module. We know that the category of 𝒪X\mathcal{O}_X -modules on XX has enough injectives, see Injectives, Lemma . Hence we can choose an injective resolution ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet . As is customary we define H^i(X, \mathcal{F}) = H^i(\Gamma(X, \mathcal{I}^\bullet)) to be the ii th cohomology group of ℱ\mathcal{F} . The family of functors Hi(X,−)H^i(X, -) forms a universal δ\delta -functor from Mod(𝒪X)→Mod𝒪X(X)\textit{Mod}(\mathcal{O}_X) \to \text{Mod}_{\mathcal{O}_X(X)} . Let f:(X,𝒪X)→(Y,𝒪Y)f : (X, \mathcal{O}_X) \to (Y, \mathcal{O}_Y) be a morphism of ringed spaces. With ℱ[0]→ℐ•\mathcal{F}[0] \to \mathcal{I}^\bullet as above we define R^if_*\mathcal{F} = H^i(f_*\mathcal{I}^\bullet) to be the ii th higher direct image of ℱ\mathcal{F} . The family of functors Rif*R^if_* forms a universal δ\delta -functor from Mod(𝒪X)→Mod(𝒪Y)\textit{Mod}(\mathcal{O}_X) \to \textit{Mod}(\mathcal{O}_Y) .

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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
  "fragmentKind": "section",
  "id": "urn:stacks:clir:sheaf-cohomology#ST_01DZ",
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      "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 $\\textit{Mod}(\\mathcal{O}_X) \\to \\textit{Mod}(\\mathcal{O}_Y)$."
    }
  ]
}

tag/0716

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Derived functors

We briefly explain how to get right derived functors using resolution functors. For the unbounded derived functors, please see Section . Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is abelian, see Modules, Lemma . In this chapter we will write

K(𝒪X)=K(Mod(𝒪X))andD(𝒪X)=D(Mod(𝒪X)).K(\mathcal{O}_X) = K(\textit{Mod}(\mathcal{O}_X)) \quad \text{and} \quad D(\mathcal{O}_X) = D(\textit{Mod}(\mathcal{O}_X)).

and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition and Definition . By Derived Categories, Remark there exists a resolution functor

j=jX:K+(Mod(𝒪X))⟶K+(ℐ)j = j_X : K^{+}(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K^{+}(\mathcal{I})

where ℐ\mathcal{I} is the strictly full additive subcategory of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) consisting of injective sheaves. For any left exact functor F:Mod(𝒪X)→ℬF : \textit{Mod}(\mathcal{O}_X) \to \mathcal{B} into any abelian category ℬ\mathcal{B} we will denote RFRF the right derived functor described in Derived Categories, Section and constructed using the resolution functor jXj_X just described: RF = F \circ j_X' : D^{+}(X) \longrightarrow D^{+}(\mathcal{B}) see Derived Categories, Lemma for notation. Note that we may think of RFRF as defined on Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) , Comp+(Mod(𝒪X))\text{Comp}^{+}(\textit{Mod}(\mathcal{O}_X)) , K+(X)K^{+}(X) , or D+(X)D^{+}(X) depending on the situation. According to Derived Categories, Definition we obtain the ii th right derived functor R^iF = H^i \circ RF : Mod (\mathcal{O}_X) \longrightarrow \mathcal{B} so that R0F=FR^0F = F and {RiF,δ}i≥0\{R^iF, \delta\}_{i \geq 0} is universal δ\delta -functor, see Derived Categories, Lemma . Here are two special cases of this construction. Given a ring RR we write K(R)=K(ModR)K(R) = K(\text{Mod}_R) and D(R)=D(ModR)D(R) = D(\text{Mod}_R) and similarly for bounded versions. For any open U⊂XU \subset X we have a left exact functor Γ(U,−):Mod(𝒪X)⟶Mod𝒪X(U)\Gamma(U, -) : \textit{Mod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)} which gives rise to R\Gamma(U, -) : D^{+}(X) \longrightarrow D^{+}(\mathcal{O}_X(U)) by the discussion above. We set Hi(U,−)=RiΓ(U,−)H^i(U, -) = R^i\Gamma(U, -) . If U=XU = X we recover (). If f:X→Yf : X \to Y is a morphism of ringed spaces, then we have the left exact functor f*:Mod(𝒪X)⟶Mod(𝒪Y)f_* : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_Y) which gives rise to the derived pushforward Rf_* : D^{+}(X) \longrightarrow D^{+}(Y) The ii th cohomology sheaf of Rf*ℱ•Rf_*\mathcal{F}^\bullet is denoted Rif*ℱ•R^if_*\mathcal{F}^\bullet and called the ii th higher direct image in accordance with (). The two displayed functors above are exact functors of derived categories. Abuse of notation: When the functor Rf*Rf_* , or any other derived functor, is applied to a sheaf ℱ\mathcal{F} on XX or a complex of sheaves it is understood that ℱ\mathcal{F} has been replaced by a suitable resolution of ℱ\mathcal{F} . To facilitate this kind of operation we will say, given an object ℱ•∈D(𝒪X)\mathcal{F}^\bullet \in D(\mathcal{O}_X) , that a bounded below complex ℐ•\mathcal{I}^\bullet of injectives of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) represents ℱ•\mathcal{F}^\bullet in the derived category if there exists a quasi-isomorphism ℱ•→ℐ•\mathcal{F}^\bullet \to \mathcal{I}^\bullet . In the same vein the phrase ``let α:ℱ•→𝒢•\alpha : \mathcal{F}^\bullet \to \mathcal{G}^\bullet be a morphism of D(𝒪X)D(\mathcal{O}_X) '' does not mean that α\alpha is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.

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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
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    {
      "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."
    }
  ]
}

tag/09SW

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space. We say a presheaf of sets ℱ\mathcal{F} is flasque or flabby if for every U⊂VU \subset V open in XX the restriction map ℱ(V)→ℱ(U)\mathcal{F}(V) \to \mathcal{F}(U) is surjective.

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  "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",
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      "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}"
    }
  ]
}

tag/09SY

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. Any flasque 𝒪X\mathcal{O}_X -module is acyclic for RΓ(X,−)R\Gamma(X, -) as well as RΓ(U,−)R\Gamma(U, -) for any open UU of XX .

Original data · JSON
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  "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}"
    }
  ]
}

tag/0FKS

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. For every sheaf of 𝒪X\mathcal{O}_X -modules ℱ\mathcal{F} there is a resolution

0→ℱ→f*f*ℱ→f*f*f*f*ℱ→f*f*f*f*f*f*ℱ→…0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots

functorial in ℱ\mathcal{F} such that each term f*f*…f*f*ℱf_*f^* \ldots f_*f^*\mathcal{F} is a flasque 𝒪X\mathcal{O}_X -module and such that for all x∈Xx \in X the map

ℱx[0]→((f*f*ℱ)x→(f*f*f*f*ℱ)x→(f*f*f*f*f*f*ℱ)x→…)\mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big)

is a homotopy equivalence in the category of complexes of 𝒪X,x\mathcal{O}_{X, x} -modules.

Original data · JSON
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{
  "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}"
    }
  ]
}

tag/0109

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

A category 𝒜\mathcal{A} is abelian if it is additive, if all kernels and cokernels exist, and if the natural map Coim(f)→Im(f)\Coim(f) \to \Im(f) is an isomorphism for all morphisms ff of 𝒜\mathcal{A} .

Original data · JSON
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  "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"
}

tag/010N

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} and ℬ\mathcal{B} be abelian categories. Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be a functor.

  • If FF is either left or right exact, then it is additive.

  • FF is left exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence 0→F(A)→F(B)→F(C)0 \to F(A) \to F(B) \to F(C) is exact.

  • FF is right exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence F(A)→F(B)→F(C)→0F(A) \to F(B) \to F(C) \to 0 is exact.

  • FF is exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence 0→F(A)→F(B)→F(C)→00 \to F(A) \to F(B) \to F(C) \to 0 is exact.

Original data · JSON
JSONRead only
{
  "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"
}

tag/013K

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category. Assume 𝒜\mathcal{A} has enough injectives.

  • Any object of 𝒜\mathcal{A} has an injective resolution.

  • If Hn(K•)=0H^n(K^\bullet) = 0 for all n≪0n \ll 0 then K•K^\bullet has an injective resolution.

  • If K•K^\bullet is a complex with Kn=0K^n = 0 for n<an < a , then there exists an injective resolution α:K•→I•\alpha : K^\bullet \to I^\bullet with In=0I^n = 0 for n<an < a such that each αn:Kn→In\alpha^n : K^n \to I^n is injective.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/015B

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category with enough injectives. Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be a left exact functor.

  • For any short exact sequence 0→A•→B•→C•→00 \to A^\bullet \to B^\bullet \to C^\bullet \to 0 of complexes in Comp+(𝒜)\text{Comp}^{+}(\mathcal{A}) there is an associated long exact sequence

    …→Hi(RF(A•))→Hi(RF(B•))→Hi(RF(C•))→Hi+1(RF(A•))→…\ldots \to H^i(RF(A^\bullet)) \to H^i(RF(B^\bullet)) \to H^i(RF(C^\bullet)) \to H^{i + 1}(RF(A^\bullet)) \to \ldots

  • The functors RiF:𝒜→ℬR^iF : \mathcal{A} \to \mathcal{B} are zero for i<0i < 0 . Also R0F=F:𝒜→ℬR^0F = F : \mathcal{A} \to \mathcal{B} .

  • We have RiF(I)=0R^iF(I) = 0 for i>0i > 0 and II injective.

  • The sequence (RiF,δ)(R^iF, \delta) forms a universal δ\delta -functor (see Homology, Definition ) from 𝒜\mathcal{A} to ℬ\mathcal{B} .

Original data · JSON
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{
  "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}"
    }
  ]
}

tag/015C

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be an additive functor between abelian categories and assume RF:D+(𝒜)→D+(ℬ)RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) is everywhere defined. Let AA be an object of 𝒜\mathcal{A} .

  • AA is right acyclic for FF if and only if F(A)→R0F(A)F(A) \to R^0F(A) is an isomorphism and RiF(A)=0R^iF(A) = 0 for all i>0i > 0 ,

  • if FF is left exact, then AA is right acyclic for FF if and only if RiF(A)=0R^iF(A) = 0 for all i>0i > 0 .

Original data · JSON
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  "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",
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    {
      "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}"
    }
  ]
}

tag/05SV

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

In Situation . We say FF is right derivable , or that RFRF everywhere defined if RFRF is defined at every object of 𝒟\mathcal{D} . We say FF is left derivable , or that LFLF everywhere defined if LFLF is defined at every object of 𝒟\mathcal{D} .

Original data · JSON
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{
  "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}"
    }
  ]
}

tag/05T4

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Here F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} is an additive functor between abelian categories. This induces exact functors

F:K(𝒜)→K(ℬ),K+(𝒜)→K+(ℬ),K−(𝒜)→K−(ℬ).F : K(\mathcal{A}) \to K(\mathcal{B}), \quad K^{+}(\mathcal{A}) \to K^{+}(\mathcal{B}), \quad K^{-}(\mathcal{A}) \to K^{-}(\mathcal{B}).

See Lemma . We also denote FF the composition K(𝒜)→D(ℬ)K(\mathcal{A}) \to D(\mathcal{B}) , K+(𝒜)→D+(ℬ)K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) , and K−(𝒜)→D−(ℬ)K^{-}(\mathcal{A}) \to D^-(\mathcal{B}) of FF with the localization functor K(ℬ)→D(ℬ)K(\mathcal{B}) \to D(\mathcal{B}) , etc. This situation leads to four derived functors we will consider in the following.

  • The right derived functor of F:K(𝒜)→D(ℬ)F : K(\mathcal{A}) \to D(\mathcal{B}) relative to the multiplicative system Qis(𝒜)\text{Qis}(\mathcal{A}) .

  • The right derived functor of F:K+(𝒜)→D+(ℬ)F : K^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) relative to the multiplicative system Qis+(𝒜)\text{Qis}^{+}(\mathcal{A}) .

  • The left derived functor of F:K(𝒜)→D(ℬ)F : K(\mathcal{A}) \to D(\mathcal{B}) relative to the multiplicative system Qis(𝒜)\text{Qis}(\mathcal{A}) .

  • The left derived functor of F:K−(𝒜)→D−(ℬ)F : K^{-}(\mathcal{A}) \to D^{-}(\mathcal{B}) relative to the multiplicative system Qis−(𝒜)\text{Qis}^-(\mathcal{A}) . Each of these cases is an example of Situation .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/05TD

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be an additive functor between abelian categories and assume RF:D+(𝒜)→D+(ℬ)RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) is everywhere defined.

  • We have RiF=0R^iF = 0 for i<0i < 0 ,

  • R0FR^0F is left exact,

  • the map F→R0FF \to R^0F is an isomorphism if and only if FF is left exact.

Original data · JSON
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  "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": [
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      "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}"
    }
  ]
}

tag/05TE

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be an additive functor between abelian categories and assume RF:D+(𝒜)→D+(ℬ)RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) is everywhere defined.

  • The functors RiFR^iF , i≥0i \geq 0 come equipped with a canonical structure of a δ\delta -functor from 𝒜→ℬ\mathcal{A} \to \mathcal{B} , see Homology, Definition .

  • If every object of 𝒜\mathcal{A} is a subobject of a right acyclic object for FF , then {RiF,δ}i≥0\{R^iF, \delta\}_{i \geq 0} is a universal δ\delta -functor, see Homology, Definition .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/05TI

Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category with enough injectives.

  • For any exact functor F:K+(𝒜)→𝒟F : K^{+}(\mathcal{A}) \to \mathcal{D} into a triangulated category 𝒟\mathcal{D} the right derived functor

    RF:D+(𝒜)⟶𝒟RF : D^{+}(\mathcal{A}) \longrightarrow \mathcal{D}

    is everywhere defined.

  • For any additive functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} into an abelian category ℬ\mathcal{B} the right derived functor

    RF:D+(𝒜)⟶D+(ℬ)RF : D^{+}(\mathcal{A}) \longrightarrow D^{+}(\mathcal{B})

    is everywhere defined.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

Packages in the snapshot

  • Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
  • 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
cohomology_vanishes_positive: TRUE_ONLY — установлено Выведено 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) …и ещё 20 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены 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 Ответ 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) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (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…) Вместе 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 proof-граф: 44 узлов — поле evaluation готово для law_explain

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JSON · calculations, sources and exact data

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{
  "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"
}
evaluation SHA-256
sha256:7b39be0f6824c88aa4f9fda1cf5aca51eaf6f428c57cf06e55d13007ad10c3a4
Original data · JSON
JSONRead only
{
  "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
}

Вялость — достаточное условие ацикличности: в положительных степенях когомологий нет.

Condition

Инъективный пучок вял

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

09SW: the sheaf is flasque (flabby)

f

Input facts

  • the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: abxx: x

  • the category is AbAb , the category of abelian groups

    ab: ab

  • the functor is Γ(X,−):Ab(X)→Ab,F↦Γ(X,F)=F(X)Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)

    g: gammax: x

  • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: fx: x

  • each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: f

  • 006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: f

  • 006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: f

  • for every injection A↪BA ↪ B and every morphism A→JA → J there exists a morphism B→JB → J making the diagram commute

    j: f

Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "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
}
Why this resultApplied rules and conditions

Derivation path13 steps

  1. 1

    the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: urn:case:stacks:sh:abx; x: urn:case:stacks:sh:x

    case fact
  2. 2

    FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    case fact
  3. 3

    each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: urn:case:stacks:sh:f

    case fact
  4. 4

    006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006K

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
    rule
  5. 5

    006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: urn:case:stacks:sh:f

    case fact
  6. 6

    006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: urn:case:stacks:sh:f

    case fact
  7. 7

    006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    FF is a sheaf of sets on XX: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing
    rule
  8. 8

    0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 0070

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition
    rule
  9. 9

    0070 (2): an abelian sheaf on XX is an object of Ab(X)Ab(X)

    object of: urn:case:stacks:sh:f, urn:case:stacks:sh:abx

    tag 0070

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX
    rule
  10. 10

    for every injection A↪BA ↪ B and every morphism A→JA → J there exists a morphism B→JB → J making the diagram commute

    j: urn:case:stacks:sh:f

    case fact
  11. 11

    0135: an object JJ is injective if every morphism A→JA → J extends along every injection A↪BA ↪ B

    the object is injective: j: urn:case:stacks:sh:f

    tag 0135

    Identifier
    urn:stacks:clir:categories#InjectiveByLifting
    rule
  12. 12

    09SX with 01AD: an injective object of Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is flasque

    09SW: the sheaf is flasque (flabby): f: urn:case:stacks:sh:f

    tag 09SX, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque
    rule
  13. 13

    Query evaluation

    query

verified by the engine: 7 · case fact: 6 · Full graph: 36 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

  • 0135: an object JJ is injective if every morphism A→JA → J extends along every injection A↪BA ↪ B

    Identifier
    urn:stacks:clir:categories#InjectiveByLifting
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition

  • 0070 (2): an abelian sheaf on XX is an object of Ab(X)Ab(X)

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX

  • 09SX with 01AD: an injective object of Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is flasque

    Identifier
    urn:stacks:clir:sheaf-cohomology#InjectiveSheafIsFlasque

  • 006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
Other rules in the evaluation16

Applied in the overall evaluation, but not on the proof path for this answer.

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

  • 010N (2): FF is left exact if for every short exact sequence 0→A→B→C→00 → A → B → C → 0 the sequence 0→F(A)→F(B)→F(C)0 → F(A) → F(B) → F(C) is exact

    Identifier
    urn:stacks:clir:categories#LeftExactBySES

  • 010N (1): if FF is left exact, then it is additive

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive

  • 0109: a category is abelian if it is additive, all kernels and cokernels exist, and Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all ff

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbAdditive

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCoimageImage

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCokernels

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbKernels

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXAdditive

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for every morphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCoimageImage

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCokernels

  • 01DG: the category of abelian sheaves on a topological space XX has enough injectives

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXKernels

  • 09SY with 01AD: a flasque abelian sheaf is right acyclic for Γ(X,−)Γ(X, −)

    Identifier
    urn:stacks:clir:sheaf-cohomology#FlasqueIsAcyclic

  • 0716: Γ(X,−)Γ(X, −) is a functor from Ab(X)Ab(X) to AbAb

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween

  • 0716: Γ(X,−)Γ(X, −) is a left exact functor

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact

  • 0FKS with 01AD: every abelian sheaf FF on XX has the Godement resolution 0→F→f*f*F→…0 → F → f_*f^*F → … by flasque sheaves

    Identifier
    urn:stacks:clir:sheaf-cohomology#GodementResolutionExists

Derived result for this query

  • 09SW: the sheaf is flasque (flabby)

    f: f
Other derived facts7

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: fx: x

  • FF is a sheaf of sets on XX

    f: fx: x

  • 0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf

    f: fx: x
  • Original data · JSON
    JSONRead only
    "object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx)"

  • 0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)

    f: f

  • the object is injective

    j: f
  • Original data · JSON
    JSONRead only
    "right_acyclic_for(urn:case:stacks:sh:f, urn:case:stacks:sh:gamma)"
006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms
fx
urn:case:stacks:sh:furn:case:stacks:sh:x
FF is a sheaf of sets on XX
fx
urn:case:stacks:sh:furn:case:stacks:sh:x
0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf
fx
urn:case:stacks:sh:furn:case:stacks:sh:x
0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)
f
urn:case:stacks:sh:f
the object is injective
j
urn:case:stacks:sh:f
09SW: the sheaf is flasque (flabby)
f
urn:case:stacks:sh:f

15 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

  1. 1

    006T: a presheaf refuted by an open covering is not a sheaf of sets on XX

    What is missing

    • FF is not a sheaf on XX : the sheaf condition 006T fails on some open coveringf, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering
    rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 6 layer
query_evaluationflasquerule_applicationInjectiveSheafIsFlasquerule_applicationAbelianSheafIsObjectOfAbXrule_applicationInjectiveByLiftingassertionsheaves_categoryrule_applicationAbelianSheafByDefinitionassertionlifting_propertyrule_applicationSheafByGluingrule_applicationabelian_presheaf_on/sufficientassertionpresheaf_of_sets_onassertioncompatible_sections_glueassertiongluing_is_uniqueassertionabelian_group_structure

Proof nodes: 36 · assertion 8, rule_application 23, constraint_check 4, query_evaluation 1

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

Download JSON ↓
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON

This block is too large for inline viewing. It is included in full in the document JSON, without truncation.

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SourcesExcerpts: 13

tag/006K

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A presheaf of abelian groups on XX or an abelian presheaf over XX is a presheaf of sets ℱ\mathcal{F} such that for each open U⊂XU \subset X the set ℱ(U)\mathcal{F}(U) is endowed with the structure of an abelian group, and such that all restriction maps ρVU\rho^U_V are homomorphisms of abelian groups, see Lemma above.

  • A morphism of abelian presheaves over XX φ:ℱ→𝒢\varphi : \mathcal{F} \to \mathcal{G} is a morphism of presheaves of sets which induces a homomorphism of abelian groups ℱ(U)→𝒢(U)\mathcal{F}(U) \to \mathcal{G}(U) for every open U⊂XU \subset X .

  • The category of presheaves of abelian groups on XX is denoted PAb(X)\textit{PAb}(X) .

Original data · JSON
JSONRead only
{
  "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",
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    {
      "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}"
    }
  ]
}

tag/006T

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A sheaf ℱ\mathcal{F} of sets on XX is a presheaf of sets which satisfies the following additional property: Given any open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i and any collection of sections si∈ℱ(Ui)s_i \in \mathcal{F}(U_i) , i∈Ii \in I such that ∀i,j∈I\forall i, j\in I

    si|Ui∩Uj=sj|Ui∩Ujs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}

    there exists a unique section s∈ℱ(U)s \in \mathcal{F}(U) such that si=s|Uis_i = s|_{U_i} for all i∈Ii \in I .

  • A morphism of sheaves of sets is simply a morphism of presheaves of sets.

  • The category of sheaves of sets on XX is denoted Sh(X)\Sh(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0070

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • An abelian sheaf on XX or sheaf of abelian groups on XX is an abelian presheaf on XX such that the underlying presheaf of sets is a sheaf.

  • The category of sheaves of abelian groups is denoted Ab(X)\textit{Ab}(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/01AD

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Introduction

In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of 𝐙―\underline{\mathbf{Z}} -modules. Basic references are , and . We work out what happens for sheaves of modules on ringed topoi in another chapter (see Modules on Sites, Section ), although there we will mostly just duplicate the discussion from this chapter.

Original data · JSON
JSONRead only
{
  "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."
    }
  ]
}

tag/01AG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is an abelian category. Moreover a complex

ℱ→𝒢→ℋ\mathcal{F} \to \mathcal{G} \to \mathcal{H}

is exact at 𝒢\mathcal{G} if and only if for all x∈Xx \in X the complex

ℱx→𝒢x→ℋx\mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x

is exact at 𝒢x\mathcal{G}_x .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/01DG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space. The category of abelian sheaves on XX has enough injectives. In fact it has functorial injective embeddings.

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0716

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Derived functors

We briefly explain how to get right derived functors using resolution functors. For the unbounded derived functors, please see Section . Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is abelian, see Modules, Lemma . In this chapter we will write

K(𝒪X)=K(Mod(𝒪X))andD(𝒪X)=D(Mod(𝒪X)).K(\mathcal{O}_X) = K(\textit{Mod}(\mathcal{O}_X)) \quad \text{and} \quad D(\mathcal{O}_X) = D(\textit{Mod}(\mathcal{O}_X)).

and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition and Definition . By Derived Categories, Remark there exists a resolution functor

j=jX:K+(Mod(𝒪X))⟶K+(ℐ)j = j_X : K^{+}(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K^{+}(\mathcal{I})

where ℐ\mathcal{I} is the strictly full additive subcategory of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) consisting of injective sheaves. For any left exact functor F:Mod(𝒪X)→ℬF : \textit{Mod}(\mathcal{O}_X) \to \mathcal{B} into any abelian category ℬ\mathcal{B} we will denote RFRF the right derived functor described in Derived Categories, Section and constructed using the resolution functor jXj_X just described: RF = F \circ j_X' : D^{+}(X) \longrightarrow D^{+}(\mathcal{B}) see Derived Categories, Lemma for notation. Note that we may think of RFRF as defined on Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) , Comp+(Mod(𝒪X))\text{Comp}^{+}(\textit{Mod}(\mathcal{O}_X)) , K+(X)K^{+}(X) , or D+(X)D^{+}(X) depending on the situation. According to Derived Categories, Definition we obtain the ii th right derived functor R^iF = H^i \circ RF : Mod (\mathcal{O}_X) \longrightarrow \mathcal{B} so that R0F=FR^0F = F and {RiF,δ}i≥0\{R^iF, \delta\}_{i \geq 0} is universal δ\delta -functor, see Derived Categories, Lemma . Here are two special cases of this construction. Given a ring RR we write K(R)=K(ModR)K(R) = K(\text{Mod}_R) and D(R)=D(ModR)D(R) = D(\text{Mod}_R) and similarly for bounded versions. For any open U⊂XU \subset X we have a left exact functor Γ(U,−):Mod(𝒪X)⟶Mod𝒪X(U)\Gamma(U, -) : \textit{Mod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)} which gives rise to R\Gamma(U, -) : D^{+}(X) \longrightarrow D^{+}(\mathcal{O}_X(U)) by the discussion above. We set Hi(U,−)=RiΓ(U,−)H^i(U, -) = R^i\Gamma(U, -) . If U=XU = X we recover (). If f:X→Yf : X \to Y is a morphism of ringed spaces, then we have the left exact functor f*:Mod(𝒪X)⟶Mod(𝒪Y)f_* : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_Y) which gives rise to the derived pushforward Rf_* : D^{+}(X) \longrightarrow D^{+}(Y) The ii th cohomology sheaf of Rf*ℱ•Rf_*\mathcal{F}^\bullet is denoted Rif*ℱ•R^if_*\mathcal{F}^\bullet and called the ii th higher direct image in accordance with (). The two displayed functors above are exact functors of derived categories. Abuse of notation: When the functor Rf*Rf_* , or any other derived functor, is applied to a sheaf ℱ\mathcal{F} on XX or a complex of sheaves it is understood that ℱ\mathcal{F} has been replaced by a suitable resolution of ℱ\mathcal{F} . To facilitate this kind of operation we will say, given an object ℱ•∈D(𝒪X)\mathcal{F}^\bullet \in D(\mathcal{O}_X) , that a bounded below complex ℐ•\mathcal{I}^\bullet of injectives of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) represents ℱ•\mathcal{F}^\bullet in the derived category if there exists a quasi-isomorphism ℱ•→ℐ•\mathcal{F}^\bullet \to \mathcal{I}^\bullet . In the same vein the phrase ``let α:ℱ•→𝒢•\alpha : \mathcal{F}^\bullet \to \mathcal{G}^\bullet be a morphism of D(𝒪X)D(\mathcal{O}_X) '' does not mean that α\alpha is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.

Original data · JSON
JSONRead only
{
  "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",
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    {
      "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."
    }
  ]
}

tag/09SX

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. Then any injective 𝒪X\mathcal{O}_X -module is flasque.

Original data · JSON
JSONRead only
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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:sheaf-cohomology#ST_09SX",
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      "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}"
    }
  ]
}

tag/09SY

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. Any flasque 𝒪X\mathcal{O}_X -module is acyclic for RΓ(X,−)R\Gamma(X, -) as well as RΓ(U,−)R\Gamma(U, -) for any open UU of XX .

Original data · JSON
JSONRead only
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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:sheaf-cohomology#ST_09SY",
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      "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}"
    }
  ]
}

tag/0FKS

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. For every sheaf of 𝒪X\mathcal{O}_X -modules ℱ\mathcal{F} there is a resolution

0→ℱ→f*f*ℱ→f*f*f*f*ℱ→f*f*f*f*f*f*ℱ→…0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots

functorial in ℱ\mathcal{F} such that each term f*f*…f*f*ℱf_*f^* \ldots f_*f^*\mathcal{F} is a flasque 𝒪X\mathcal{O}_X -module and such that for all x∈Xx \in X the map

ℱx[0]→((f*f*ℱ)x→(f*f*f*f*ℱ)x→(f*f*f*f*f*f*ℱ)x→…)\mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big)

is a homotopy equivalence in the category of complexes of 𝒪X,x\mathcal{O}_{X, x} -modules.

Original data · JSON
JSONRead only
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  "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",
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    {
      "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}"
    }
  ]
}

tag/0109

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

A category 𝒜\mathcal{A} is abelian if it is additive, if all kernels and cokernels exist, and if the natural map Coim(f)→Im(f)\Coim(f) \to \Im(f) is an isomorphism for all morphisms ff of 𝒜\mathcal{A} .

Original data · JSON
JSONRead only
{
  "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"
}

tag/010N

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} and ℬ\mathcal{B} be abelian categories. Let F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} be a functor.

  • If FF is either left or right exact, then it is additive.

  • FF is left exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence 0→F(A)→F(B)→F(C)0 \to F(A) \to F(B) \to F(C) is exact.

  • FF is right exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence F(A)→F(B)→F(C)→0F(A) \to F(B) \to F(C) \to 0 is exact.

  • FF is exact if and only if for every short exact sequence 0→A→B→C→00 \to A \to B \to C \to 0 the sequence 0→F(A)→F(B)→F(C)→00 \to F(A) \to F(B) \to F(C) \to 0 is exact.

Original data · JSON
JSONRead only
{
  "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"
}

tag/0135

Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина

Let 𝒜\mathcal{A} be an abelian category. An object J∈Ob(𝒜)J \in \Ob(\mathcal{A}) is called injective if for every injection A↪BA \hookrightarrow B and every morphism A→JA \to J there exists a morphism B→JB \to J making the following diagram commute

\xymatrix{
A \ar[r] \ar[d] & B \ar@{-->}[ld] \\
J &
}
Диаграмма: исходный TeX
Original data · JSON
JSONRead only
{
  "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"
}

Packages in the snapshot

  • Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
  • Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
flasque: TRUE_ONLY — установлено Выведено 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) …и ещё 15 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены 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 Ответ 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) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (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…) Вместе с актами: Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина proof-граф: 36 узлов — поле evaluation готово для law_explain

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        "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",
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      "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"
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      "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"
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evaluation SHA-256
sha256:de1183bf01a85c57704d5ef59e996ed95975546c138645cf0aa651373ac6100e
Original data · JSON
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        "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
}

Инъективность влечёт вялость, поэтому инъективные резольвенты состоят из ацикличных пучков.

Condition

Вялая резольвента существует

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)

f

Input facts

  • the category is Ab(X)Ab(X) , the category of sheaves of abelian groups on XX

    c: abxx: x

  • the category is AbAb , the category of abelian groups

    ab: ab

  • the functor is Γ(X,−):Ab(X)→Ab,F↦Γ(X,F)=F(X)Γ(X, −): Ab(X) → Ab, F ↦ Γ(X, F) = F(X)

    g: gammax: x

  • FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: fx: x

  • each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: f

  • 006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: f

  • 006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: f

Package: Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "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
}
Why this resultApplied rules and conditions

Derivation path9 steps

  1. 1

    FF is a presheaf of sets on XX : a rule assigning a set F(U)F(U) to each open UU and restriction maps ρVUρ^U_V to inclusions V⊂UV ⊂ U with ρUU=idρ^U_U = id and ρWU=ρWV∘ρVUρ^U_W = ρ^V_W ∘ ρ^U_V

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    case fact
  2. 2

    each F(U)F(U) carries the structure of an abelian group and all restriction maps are homomorphisms of abelian groups

    f: urn:case:stacks:sh:f

    case fact
  3. 3

    006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006K

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
    rule
  4. 4

    006T, existence: for every open covering U=∪UiU = ∪ U_i and sections si∈F(Ui)s_i ∈ F(U_i) with si|Ui∩Uj=sj|Ui∩Ujs_i|U_i∩U_j = s_j|U_i∩U_j there exists s∈F(U)s ∈ F(U) with s|Ui=sis|U_i = s_i

    f: urn:case:stacks:sh:f

    case fact
  5. 5

    006T, uniqueness: a section s∈F(U)s ∈ F(U) is determined by its restrictions s|Uis|U_i to an open covering

    f: urn:case:stacks:sh:f

    case fact
  6. 6

    006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    FF is a sheaf of sets on XX: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing
    rule
  7. 7

    0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf: f: urn:case:stacks:sh:f; x: urn:case:stacks:sh:x

    tag 0070

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition
    rule
  8. 8

    0FKS with 01AD: every abelian sheaf FF on XX has the Godement resolution 0→F→f*f*F→…0 → F → f_*f^*F → … by flasque sheaves

    0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution): f: urn:case:stacks:sh:f

    tag 0FKS, tag 01AD

    Identifier
    urn:stacks:clir:sheaf-cohomology#GodementResolutionExists
    rule
  9. 9

    Query evaluation

    query

verified by the engine: 5 · case fact: 4 · Full graph: 26 nodes

Steps of the saved proof from the case facts to the answer. Formulas are shown as written in the norm with bound values substituted; the page recomputes nothing.

Basis of this answer

Rules on the saved proof path for this answer.

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 0070 (1): an abelian presheaf on XX whose underlying presheaf of sets is a sheaf is an abelian sheaf

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafByDefinition

  • 0FKS with 01AD: every abelian sheaf FF on XX has the Godement resolution 0→F→f*f*F→…0 → F → f_*f^*F → … by flasque sheaves

    Identifier
    urn:stacks:clir:sheaf-cohomology#GodementResolutionExists

  • 006T: a presheaf of sets is a sheaf if compatible families of sections over any open covering glue to a unique section

    Identifier
    urn:stacks:clir:sheaf-cohomology#SheafByGluing

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    Identifier
    urn:stacks:clir:sheaf-cohomology#abelian_presheaf_on/sufficient
Other rules in the evaluation12

Applied in the overall evaluation, but not on the proof path for this answer.

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbAdditive

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCoimageImage

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbCokernels

  • 01AG on the one-point space: Ab=Mod(Z)Ab = Mod(Z) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbKernels

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular additive

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXAdditive

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for every morphism

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCoimageImage

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all cokernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXCokernels

  • 01DG: the category of abelian sheaves on a topological space XX has enough injectives

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXEnoughInjectives

  • 01AG with 01AD: Ab(X)=Mod(ZX)Ab(X) = Mod(Z_X) is abelian, in particular all kernels exist

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbXKernels

  • 0070 (2): an abelian sheaf on XX is an object of Ab(X)Ab(X)

    Identifier
    urn:stacks:clir:sheaf-cohomology#AbelianSheafIsObjectOfAbX

  • 0716: Γ(X,−)Γ(X, −) is a functor from Ab(X)Ab(X) to AbAb

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsBetween

  • 0716: Γ(X,−)Γ(X, −) is a left exact functor

    Identifier
    urn:stacks:clir:sheaf-cohomology#GlobalSectionsLeftExact

Derived result for this query

  • 0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)

    f: f
Other derived facts4

  • 006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms

    f: fx: x

  • FF is a sheaf of sets on XX

    f: fx: x

  • 0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf

    f: fx: x
  • Original data · JSON
    JSONRead only
    "object_of(urn:case:stacks:sh:f, urn:case:stacks:sh:abx)"
006K: an abelian presheaf on XX is a presheaf of sets FF such that each F(U)F(U) is an abelian group and all restriction maps are group homomorphisms
fx
urn:case:stacks:sh:furn:case:stacks:sh:x
FF is a sheaf of sets on XX
fx
urn:case:stacks:sh:furn:case:stacks:sh:x
0070: FF is an abelian sheaf on XX — an abelian presheaf whose underlying presheaf of sets is a sheaf
fx
urn:case:stacks:sh:furn:case:stacks:sh:x
0FKS: the sheaf has a functorial resolution by flasque sheaves (the Godement resolution)
f
urn:case:stacks:sh:f

11 further derived facts are not shown: the engine keeps the ones relevant to the question in its compact answer. The full list is in the calculation JSON below.

What could defeat the conclusion1 rules

  1. 1

    006T: a presheaf refuted by an open covering is not a sheaf of sets on XX

    What is missing

    • FF is not a sheaf on XX : the sheaf condition 006T fails on some open coveringf, xNot establishedthis is the missing one

    Source: tag 006T

    Identifier
    urn:stacks:clir:sheaf-cohomology#NotSheafByRefutingCovering
    rule

These are the rules whose head answers the question, with their unmet premises. A missing fact is not a refuted one.

Proof graph

Proof graph · 5 layer
query_evaluationhas_flasque_resolutionrule_applicationGodementResolutionExistsrule_applicationAbelianSheafByDefinitionrule_applicationSheafByGluingrule_applicationabelian_presheaf_on/sufficientassertionpresheaf_of_sets_onassertioncompatible_sections_glueassertiongluing_is_uniqueassertionabelian_group_structure

Proof nodes: 26 · assertion 7, rule_application 16, constraint_check 2, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    abx
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
sheaves_category
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AbXAdditive:8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507
attributes
—
conclusion
Arguments
  • Identifier
    abx
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
8c3619a6ff45bcee9cec7b25b9b139a7abf3a95c4a67bc40472dfdefd3f07507
Type
rule_application
Premises
  • fact-1
Rule
AbXAdditive
sourceAnchors
—
substitution
v0
Identifier
abx
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:AbXCoimageImage:b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26
attributes
—
conclusion
Arguments
  • Identifier
    abx
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
b31df68bfb11faeb2a82377d6974e1fca40a3fee0825427fc41386ff0abeea26
Type
rule_application
Premises
  • fact-1
Rule
AbXCoimageImage
sourceAnchors
—
substitution
v0
Identifier
abx
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:AbXCokernels:26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82
attributes
—
conclusion
Arguments
  • Identifier
    abx
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
26546323d78d8af11bbbb84b0a9cce994612c84dec159396f1f4f6d9a5cd3b82
Type
rule_application
Premises
  • fact-1
Rule
AbXCokernels
sourceAnchors
—
substitution
v0
Identifier
abx
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:AbXEnoughInjectives:cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3
attributes
—
conclusion
Arguments
  • Identifier
    abx
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
enough_injectives
evidence
—
Identifier
cda7b58df9024f6c0fe967599e1c691de351fff175fb99af7c9f028157fbd9e3
Type
rule_application
Premises
  • fact-1
Rule
AbXEnoughInjectives
sourceAnchors
—
substitution
v0
Identifier
abx
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:AbXKernels:3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb
attributes
—
conclusion
Arguments
  • Identifier
    abx
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
3d91e64ffe3000b71f052c017378e9d5d27130742b7d16f82feaae40f5798efb
Type
rule_application
Premises
  • fact-1
Rule
AbXKernels
sourceAnchors
—
substitution
v0
Identifier
abx
Type
entity_ref
v1
Identifier
x
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    ab
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_groups_category
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AbAdditive:ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a
attributes
—
conclusion
Arguments
  • Identifier
    ab
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
ef54ddbc40bdc4ff28a4d80a90a6faafcfc0f36bf744413f221fe972eb230e5a
Type
rule_application
Premises
  • fact-2
Rule
AbAdditive
sourceAnchors
—
substitution
v0
Identifier
ab
Type
entity_ref
rule_application · urn:proof:apply:AbCoimageImage:489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda
attributes
—
conclusion
Arguments
  • Identifier
    ab
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
489c92ab18648794ad122cbc5b2793f4a0551ea9e7bd14e213ca6aca6698bcda
Type
rule_application
Premises
  • fact-2
Rule
AbCoimageImage
sourceAnchors
—
substitution
v0
Identifier
ab
Type
entity_ref
rule_application · urn:proof:apply:AbCokernels:ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527
attributes
—
conclusion
Arguments
  • Identifier
    ab
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
ff0265f8fdb930b7d4ff06ccac03c58417d86493bbb030784ccac0449b1b1527
Type
rule_application
Premises
  • fact-2
Rule
AbCokernels
sourceAnchors
—
substitution
v0
Identifier
ab
Type
entity_ref
rule_application · urn:proof:apply:AbKernels:9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61
attributes
—
conclusion
Arguments
  • Identifier
    ab
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
9ceba27ac85a6f3cc3cf70c2f872c9d19cda5237e727ac553d53ea0e90127e61
Type
rule_application
Premises
  • fact-2
Rule
AbKernels
sourceAnchors
—
substitution
v0
Identifier
ab
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    gamma
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
global_sections_functor
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:GlobalSectionsBetween:e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a
attributes
—
conclusion
Arguments
  • Identifier
    gamma
    Type
    entity_ref
  • Identifier
    abx
    Type
    entity_ref
  • Identifier
    ab
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
functor_between
evidence
—
Identifier
e7ffe1b948adf5af54d8708376f83345aec002dccfab069236cce8c73e87156a
Type
rule_application
Premises
  • fact-1
  • fact-2
  • fact-3
Rule
GlobalSectionsBetween
sourceAnchors
—
substitution
v0
Identifier
gamma
Type
entity_ref
v1
Identifier
abx
Type
entity_ref
v2
Identifier
ab
Type
entity_ref
v3
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:GlobalSectionsLeftExact:cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f
attributes
—
conclusion
Arguments
  • Identifier
    gamma
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preserves_left_exactness
evidence
—
Identifier
cbe69694c69ba24703c0c47c7653e1000e98b1f952b9f301430d49558c3cb87f
Type
rule_application
Premises
  • fact-3
Rule
GlobalSectionsLeftExact
sourceAnchors
—
substitution
v0
Identifier
gamma
Type
entity_ref
v1
Identifier
x
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-4
attributes
assertion
fact-4
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
presheaf_of_sets_on
evidence
—
Identifier
fact-4
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-5
attributes
assertion
fact-5
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_group_structure
evidence
—
Identifier
fact-5
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550
attributes
definition
concept
abelian_presheaf_on
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_presheaf_on
evidence
—
Identifier
5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550
Type
rule_application
Premises
  • fact-4
  • fact-5
Rule
abelian_presheaf_on/sufficient
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-6
attributes
assertion
fact-6
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
compatible_sections_glue
evidence
—
Identifier
fact-6
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-7
attributes
assertion
fact-7
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
gluing_is_unique
evidence
—
Identifier
fact-7
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:SheafByGluing:1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
sheaf_of_sets_on
evidence
—
Identifier
1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15
Type
rule_application
Premises
  • fact-4
  • fact-6
  • fact-7
Rule
SheafByGluing
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:AbelianSheafByDefinition:a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_sheaf_on
evidence
—
Identifier
a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b
Type
rule_application
Premises
  • 1c1b3ee5d6f76d3b8f5e71d4346fedff94104faef7eb2b49772ce02e87c9fe15
  • 5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550
Rule
AbelianSheafByDefinition
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:AbelianSheafIsObjectOfAbX:a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    abx
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
object_of
evidence
—
Identifier
a4113766b50bc85db8d50d2f5f12a167b306afe711be27f2d36df088e64657d1
Type
rule_application
Premises
  • a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b
  • fact-1
Rule
AbelianSheafIsObjectOfAbX
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
abx
Type
entity_ref
v2
Identifier
x
Type
entity_ref
rule_application · urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_flasque_resolution
evidence
—
Identifier
efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5
Type
rule_application
Premises
  • a277a096fc991e0691864c116ba6a191c497f899e5254bebdc7dbb11ad7e7a2b
Rule
GodementResolutionExists
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
constraint_check · urn:proof:constraint:DeclaredSheafNotRefutedByData:a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7
attributes
—
conclusion
constraint
DeclaredSheafNotRefutedByData
requirementStatus
Not established
Calculation status
Undetermined
triggerStatus
Satisfied
evidence
—
Identifier
a3086c1ab2d45ded5da8e0cc9af87054b331f7644b3f724323051fa23c2cbfe7
Type
constraint_check
Premises
  • fact-4
  • fact-6
  • fact-7
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd
attributes
—
conclusion
constraint
abelian_presheaf_on/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd
Type
constraint_check
Premises
  • 5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550
  • fact-4
  • fact-5
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
x
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_flasque_resolution
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "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"
            }
          }
        },
        {
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          },
          "evidence": [],
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          "sourceAnchors": []
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            "kind": "literal",
            "polarity": "positive",
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          },
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          "attributes": {},
          "conclusion": {
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              }
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            "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"
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            "v1": {
              "id": "urn:case:stacks:sh:x",
              "kind": "entity_ref"
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            "args": [
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                "id": "urn:case:stacks:sh:f",
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              {
                "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": [
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            "urn:proof:apply:abelian_presheaf_on/sufficient:5aee1f4ad9975105115e78992348b4746cacb2acabb4bb84bd75e90f5fdca550"
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            "v1": {
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          "attributes": {},
          "conclusion": {
            "args": [
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                "id": "urn:case:stacks:sh:f",
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              {
                "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"
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          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
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                "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"
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            "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"
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        },
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          "attributes": {},
          "conclusion": {
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              "args": [
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              "kind": "literal",
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              "predicate": "urn:stacks:clir:sheaf-cohomology#has_flasque_resolution"
            },
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          "evidence": [],
          "id": "urn:proof:query:mcp",
          "kind": "query_evaluation",
          "premises": [
            "urn:proof:apply:GodementResolutionExists:efebb3102f08344bb53276573ed09e2f919d28be88f1d7d8642feb028858b2b5"
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        "urn:proof:constraint:abelian_presheaf_on/necessary:69fe3a3837154e6d8c670ec6d8e8ffac94a848314f66b3d9ea2ecb4c5f7c50cd",
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  "rulesApplied": [
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    "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"
  ],
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      "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)"
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SourcesExcerpts: 8

tag/006K

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A presheaf of abelian groups on XX or an abelian presheaf over XX is a presheaf of sets ℱ\mathcal{F} such that for each open U⊂XU \subset X the set ℱ(U)\mathcal{F}(U) is endowed with the structure of an abelian group, and such that all restriction maps ρVU\rho^U_V are homomorphisms of abelian groups, see Lemma above.

  • A morphism of abelian presheaves over XX φ:ℱ→𝒢\varphi : \mathcal{F} \to \mathcal{G} is a morphism of presheaves of sets which induces a homomorphism of abelian groups ℱ(U)→𝒢(U)\mathcal{F}(U) \to \mathcal{G}(U) for every open U⊂XU \subset X .

  • The category of presheaves of abelian groups on XX is denoted PAb(X)\textit{PAb}(X) .

Original data · JSON
JSONRead only
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  "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",
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    {
      "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}"
    }
  ]
}

tag/006T

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • A sheaf ℱ\mathcal{F} of sets on XX is a presheaf of sets which satisfies the following additional property: Given any open covering U=⋃i∈IUiU = \bigcup_{i \in I} U_i and any collection of sections si∈ℱ(Ui)s_i \in \mathcal{F}(U_i) , i∈Ii \in I such that ∀i,j∈I\forall i, j\in I

    si|Ui∩Uj=sj|Ui∩Ujs_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}

    there exists a unique section s∈ℱ(U)s \in \mathcal{F}(U) such that si=s|Uis_i = s|_{U_i} for all i∈Ii \in I .

  • A morphism of sheaves of sets is simply a morphism of presheaves of sets.

  • The category of sheaves of sets on XX is denoted Sh(X)\Sh(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/0070

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space.

  • An abelian sheaf on XX or sheaf of abelian groups on XX is an abelian presheaf on XX such that the underlying presheaf of sets is a sheaf.

  • The category of sheaves of abelian groups is denoted Ab(X)\textit{Ab}(X) .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/01AD

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Introduction

In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of 𝐙―\underline{\mathbf{Z}} -modules. Basic references are , and . We work out what happens for sheaves of modules on ringed topoi in another chapter (see Modules on Sites, Section ), although there we will mostly just duplicate the discussion from this chapter.

Original data · JSON
JSONRead only
{
  "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."
    }
  ]
}

tag/01AG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is an abelian category. Moreover a complex

ℱ→𝒢→ℋ\mathcal{F} \to \mathcal{G} \to \mathcal{H}

is exact at 𝒢\mathcal{G} if and only if for all x∈Xx \in X the complex

ℱx→𝒢x→ℋx\mathcal{F}_x \to \mathcal{G}_x \to \mathcal{H}_x

is exact at 𝒢x\mathcal{G}_x .

Original data · JSON
JSONRead only
{
  "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}"
    }
  ]
}

tag/01DG

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let XX be a topological space. The category of abelian sheaves on XX has enough injectives. In fact it has functorial injective embeddings.

Original data · JSON
JSONRead only
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  "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": [
    {
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      "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}"
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tag/0716

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Derived functors

We briefly explain how to get right derived functors using resolution functors. For the unbounded derived functors, please see Section . Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. The category Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) is abelian, see Modules, Lemma . In this chapter we will write

K(𝒪X)=K(Mod(𝒪X))andD(𝒪X)=D(Mod(𝒪X)).K(\mathcal{O}_X) = K(\textit{Mod}(\mathcal{O}_X)) \quad \text{and} \quad D(\mathcal{O}_X) = D(\textit{Mod}(\mathcal{O}_X)).

and similarly for the bounded versions for the triangulated categories introduced in Derived Categories, Definition and Definition . By Derived Categories, Remark there exists a resolution functor

j=jX:K+(Mod(𝒪X))⟶K+(ℐ)j = j_X : K^{+}(\textit{Mod}(\mathcal{O}_X)) \longrightarrow K^{+}(\mathcal{I})

where ℐ\mathcal{I} is the strictly full additive subcategory of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) consisting of injective sheaves. For any left exact functor F:Mod(𝒪X)→ℬF : \textit{Mod}(\mathcal{O}_X) \to \mathcal{B} into any abelian category ℬ\mathcal{B} we will denote RFRF the right derived functor described in Derived Categories, Section and constructed using the resolution functor jXj_X just described: RF = F \circ j_X' : D^{+}(X) \longrightarrow D^{+}(\mathcal{B}) see Derived Categories, Lemma for notation. Note that we may think of RFRF as defined on Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) , Comp+(Mod(𝒪X))\text{Comp}^{+}(\textit{Mod}(\mathcal{O}_X)) , K+(X)K^{+}(X) , or D+(X)D^{+}(X) depending on the situation. According to Derived Categories, Definition we obtain the ii th right derived functor R^iF = H^i \circ RF : Mod (\mathcal{O}_X) \longrightarrow \mathcal{B} so that R0F=FR^0F = F and {RiF,δ}i≥0\{R^iF, \delta\}_{i \geq 0} is universal δ\delta -functor, see Derived Categories, Lemma . Here are two special cases of this construction. Given a ring RR we write K(R)=K(ModR)K(R) = K(\text{Mod}_R) and D(R)=D(ModR)D(R) = D(\text{Mod}_R) and similarly for bounded versions. For any open U⊂XU \subset X we have a left exact functor Γ(U,−):Mod(𝒪X)⟶Mod𝒪X(U)\Gamma(U, -) : \textit{Mod}(\mathcal{O}_X) \longrightarrow \text{Mod}_{\mathcal{O}_X(U)} which gives rise to R\Gamma(U, -) : D^{+}(X) \longrightarrow D^{+}(\mathcal{O}_X(U)) by the discussion above. We set Hi(U,−)=RiΓ(U,−)H^i(U, -) = R^i\Gamma(U, -) . If U=XU = X we recover (). If f:X→Yf : X \to Y is a morphism of ringed spaces, then we have the left exact functor f*:Mod(𝒪X)⟶Mod(𝒪Y)f_* : \textit{Mod}(\mathcal{O}_X) \longrightarrow \textit{Mod}(\mathcal{O}_Y) which gives rise to the derived pushforward Rf_* : D^{+}(X) \longrightarrow D^{+}(Y) The ii th cohomology sheaf of Rf*ℱ•Rf_*\mathcal{F}^\bullet is denoted Rif*ℱ•R^if_*\mathcal{F}^\bullet and called the ii th higher direct image in accordance with (). The two displayed functors above are exact functors of derived categories. Abuse of notation: When the functor Rf*Rf_* , or any other derived functor, is applied to a sheaf ℱ\mathcal{F} on XX or a complex of sheaves it is understood that ℱ\mathcal{F} has been replaced by a suitable resolution of ℱ\mathcal{F} . To facilitate this kind of operation we will say, given an object ℱ•∈D(𝒪X)\mathcal{F}^\bullet \in D(\mathcal{O}_X) , that a bounded below complex ℐ•\mathcal{I}^\bullet of injectives of Mod(𝒪X)\textit{Mod}(\mathcal{O}_X) represents ℱ•\mathcal{F}^\bullet in the derived category if there exists a quasi-isomorphism ℱ•→ℐ•\mathcal{F}^\bullet \to \mathcal{I}^\bullet . In the same vein the phrase ``let α:ℱ•→𝒢•\alpha : \mathcal{F}^\bullet \to \mathcal{G}^\bullet be a morphism of D(𝒪X)D(\mathcal{O}_X) '' does not mean that α\alpha is represented by a morphism of complexes. If we have an actual morphism of complexes we will say so.

Original data · JSON
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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
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      "contentHash": "sha256:f61344b89b5447a2349b548070137ca4e6f491c90033bf637527fbd9f371c876",
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      "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."
    }
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tag/0FKS

Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина

Let (X,𝒪X)(X, \mathcal{O}_X) be a ringed space. For every sheaf of 𝒪X\mathcal{O}_X -modules ℱ\mathcal{F} there is a resolution

0→ℱ→f*f*ℱ→f*f*f*f*ℱ→f*f*f*f*f*f*ℱ→…0 \to \mathcal{F} \to f_*f^*\mathcal{F} \to f_*f^*f_*f^*\mathcal{F} \to f_*f^*f_*f^*f_*f^*\mathcal{F} \to \ldots

functorial in ℱ\mathcal{F} such that each term f*f*…f*f*ℱf_*f^* \ldots f_*f^*\mathcal{F} is a flasque 𝒪X\mathcal{O}_X -module and such that for all x∈Xx \in X the map

ℱx[0]→((f*f*ℱ)x→(f*f*f*f*ℱ)x→(f*f*f*f*f*f*ℱ)x→…)\mathcal{F}_x[0] \to \Big( (f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*\mathcal{F})_x \to (f_*f^*f_*f^*f_*f^*\mathcal{F})_x \to \ldots \Big)

is a homotopy equivalence in the category of complexes of 𝒪X,x\mathcal{O}_{X, x} -modules.

Original data · JSON
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  "edition": "urn:stacks:clir:sheaf-cohomology#STACKS_COHOMOLOGY_MASTER",
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      "language": "en",
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      "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}"
    }
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}

Packages in the snapshot

  • Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
has_flasque_resolution: TRUE_ONLY — установлено Выведено 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) …и ещё 11 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: AbAdditive, AbCoimageImage, AbCokernels, AbKernels, AbXAdditive, AbXCoimageImage, AbXCokernels, AbXEnoughInjectives, AbXKernels, AbelianSheafByDefinition, AbelianSheafIsObjectOfAbX, GlobalSectionsBetween, GlobalSectionsLeftExact, GodementResolutionExists, SheafByGluing, abelian_presheaf_on/sufficient Ответ 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) (поражающее правило не сработало из-за неустановленных фактов — подайте их в facts, если они есть в деле) Право (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…) proof-граф: 26 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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    "abelian_sheaf_on(urn:case:stacks:sh:f, urn:case:stacks:sh:x)",
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        "title": "Когомологии пучков по The Stacks Project: пучок, пучковизация, H^i(X, F) как производный функтор глобальных сечений, вялые пучки — вне юрисдикции государства — доктрина"
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        "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"
}
evaluation SHA-256
sha256:22dd3ca08c675673eb855d605d37cb8e132c3aca3d914470b02f19fe7ff86530
Original data · JSON
JSONRead only
{
  "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
}

Резольвента Годемана строится для любого пучка, и вычисление когомологий всегда имеет вход.

How to cite

The snapshot is immutable: the SHA-256 of the downloadable JSON pins it, so no access date is needed.

Citation
“Почему H⁰(X, F) — это в точности глобальные сечения, отчего вялый пучок не имеет старших когомологий и всякий ли пучок допускает вялую резольвенту?”. Arxo Lens, as of 2026-09-06. https://lens.arxo.io/a/a_SaQiyajJyuLuWrrmzZd47mDW. Snapshot SHA-256: 0784573de683195255e937ea105f498e61cd96566d7d758765f5c206138128fc.
BibTeX
@misc{arxo-lens-a_SaQiyajJyuLu,
  title = {Почему H⁰(X, F) — это в точности глобальные сечения, отчего вялый пучок не имеет старших когомологий и всякий ли пучок допускает вялую резольвенту?},
  howpublished = {Arxo Lens},
  url = {https://lens.arxo.io/a/a_SaQiyajJyuLuWrrmzZd47mDW},
  note = {as of 2026-09-06; SHA-256 0784573de683195255e937ea105f498e61cd96566d7d758765f5c206138128fc}
}
Embed code

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<iframe src="https://lens.arxo.io/embed/a_SaQiyajJyuLuWrrmzZd47mDW?lang=en" width="100%" height="390" loading="lazy" title="Sheaves: H⁰ is global sections and a flasque sheaf is — Arxo Lens" style="border:0"></iframe>

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