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При каких условиях правый производный функтор определён на всей категории, когда R⁰F совпадает с F и почему инъективный объект ацикличен?

Правый производный функтор определён всюду, когда обе категории абелевы, в источнике достаточно инъективных объектов, а сам функтор аддитивен. Инъективная резольвента тогда существует у каждого объекта, R¹ считается по ней, а на инъективном объекте все старшие производные обращаются в нуль. Левая точность функтора равносильна совпадению R⁰F с F. Короткая точная последовательность порождает длинную точную последовательность производных, а в отрицательных степенях они нулевые.

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

At a glance

7

Select a result to explore its grounds

RF определён на всей категорииConditionEstablishedExecution: ComputedLogical result: EstablishedИнъективная резольвента существуетConditionEstablishedExecution: ComputedLogical result: EstablishedR¹ по инъективной резольвентеConditionEstablishedExecution: ComputedLogical result: EstablishedИнъективный объект ацикличенConditionEstablishedExecution: ComputedLogical result: EstablishedR⁰F совпадает с F при левой точностиConditionEstablishedExecution: ComputedLogical result: EstablishedДлинная точная последовательностьConditionEstablishedExecution: ComputedLogical result: EstablishedОтрицательные степени нулевыеConditionEstablishedExecution: ComputedLogical result: Established

Detailed analysis

7

Condition

RF определён на всей категории

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

f

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-b

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-b

  • all kernels exist in the category

    c: cat-b

  • all cokernels exist in the category

    c: cat-b

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-b

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

  • the functor goes from the first category to the second: F:A→BF : A → B

    f: fa: cat-ab: cat-b

  • F:Mor(x,y)→Mor(F(x),F(y))F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all objects x,yx, y

    f: f

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "homomorphism_on_hom_groups"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "rf_everywhere_defined",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path20 steps

  1. 1

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-a

    case fact
  2. 2

    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:cat-b

    case fact
  3. 3

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: urn:case:stacks:cat-a

    case fact
  4. 4

    the functor goes from the first category to the second: F:A→BF : A → B

    f: urn:case:stacks:f; a: urn:case:stacks:cat-a; b: urn:case:stacks:cat-b

    case fact
  5. 5

    F:Mor(x,y)→Mor(F(x),F(y))F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all objects x,yx, y

    f: urn:case:stacks:f

    case fact
  6. 6

    00ZY: a functor of preadditive categories is additive if it is a homomorphism of abelian groups on morphism sets

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

    tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByHomGroups
    rule
  7. 7

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-a

    case fact
  8. 8

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-a

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  9. 9

    all kernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  10. 10

    all cokernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  11. 11

    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:cat-a

    case fact
  12. 12

    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:cat-a

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  13. 13

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-b

    case fact
  14. 14

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-b

    case fact
  15. 15

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-b

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  16. 16

    all kernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  17. 17

    all cokernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  18. 18

    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:cat-b

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    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:f

    tag 05TI, tag 05SV, tag 05T4

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

    Query evaluation

    query

verified by the engine: 7 · case fact: 13 · Full graph: 24 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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 00ZY: a functor of preadditive categories is additive if it is a homomorphism of abelian groups on morphism sets

    Identifier
    urn:stacks:clir:categories#AdditiveByHomGroups

  • 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

  • 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
Other rules in the evaluation2

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

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

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

Derived result for this query

  • 05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    f: f
Other derived facts7

  • the functor is additive

    f: f

  • 0104: the category is additive

    c: cat-a

  • 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: cat-a

  • 0104: the category is additive

    c: cat-b

  • 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: cat-b

  • 05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)

    f: f

  • 05TD (1): RiF=0R^iF = 0 for i<0i < 0

    f: f
the functor is additive
f
urn:case:stacks:f
0104: the category is additive
c
urn:case:stacks:cat-a
urn:case:stacks:cat-b
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:cat-a
urn:case:stacks:cat-b
05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined
f
f
05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)
f
f
05TD (1): RiF=0R^iF = 0 for i<0i < 0
f
f

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.

Proof graph

Proof graph · 5 layer
query_evaluationrf_everywhere_definedrule_applicationRFEverywhereDefinedrule_applicationAdditiveByHomGroupsrule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientassertionenough_injectivesassertionfunctor_betweenassertionhomomorphism_on_hom_groupsrule_applicationAdditiveByFiniteProductsassertionhas_all_kernelsassertionhas_all_cokernelsassertioncoimage_to_image_isomorphismrule_applicationAdditiveByFiniteProductsassertioncoimage_to_image_isomorphismassertionhas_all_kernelsassertionhas_all_cokernelsassertionpreadditive_categoryassertionhas_finite_productsassertionpreadditive_categoryassertionhas_finite_products

Proof nodes: 24 · assertion 13, rule_application 8, constraint_check 2, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-10
attributes
assertion
fact-10
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-10
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-11
attributes
assertion
fact-11
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
enough_injectives
evidence
—
Identifier
fact-11
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-12
attributes
assertion
fact-12
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    cat-a
    Type
    entity_ref
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
functor_between
evidence
—
Identifier
fact-12
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-13
attributes
assertion
fact-13
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
homomorphism_on_hom_groups
evidence
—
Identifier
fact-13
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByHomGroups:7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_functor
evidence
—
Identifier
7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5
Type
rule_application
Premises
  • fact-13
Rule
AdditiveByHomGroups
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
attributes
—
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-4
attributes
assertion
fact-4
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-4
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-5
attributes
assertion
fact-5
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-5
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
Type
rule_application
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • fact-3
  • fact-4
  • fact-5
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-6
attributes
assertion
fact-6
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-6
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-7
attributes
assertion
fact-7
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-7
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
attributes
—
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
Type
rule_application
Premises
  • fact-6
  • fact-7
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-8
attributes
assertion
fact-8
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-8
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-9
attributes
assertion
fact-9
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-9
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
Type
rule_application
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • fact-10
  • fact-8
  • fact-9
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:RFEverywhereDefined:f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
rf_everywhere_defined
evidence
—
Identifier
f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
Type
rule_application
Premises
  • 7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
Rule
RFEverywhereDefined
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:DerivedFormDeltaFunctor:8f62e5cfbccfbc635c97b7ecd1e6ef2c33f53db9b99a642b5f0500ceabc5a9fd
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
derived_functors_form_delta_functor
evidence
—
Identifier
8f62e5cfbccfbc635c97b7ecd1e6ef2c33f53db9b99a642b5f0500ceabc5a9fd
Type
rule_application
Premises
  • f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
Rule
DerivedFormDeltaFunctor
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:NegativeDerivedVanish:9302e91a8e09a730e8bd3f7715567533ed4485085d1d6b6b087034c3cb4d8ff2
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
negative_derived_functors_vanish
evidence
—
Identifier
9302e91a8e09a730e8bd3f7715567533ed4485085d1d6b6b087034c3cb4d8ff2
Type
rule_application
Premises
  • f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
Rule
NegativeDerivedVanish
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
Type
constraint_check
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-10
  • fact-8
  • fact-9
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
Type
constraint_check
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • fact-3
  • fact-4
  • fact-5
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
rf_everywhere_defined
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "derived": [
    "additive_functor(urn:case:stacks:f)",
    "additive_category(urn:case:stacks:cat-a)",
    "abelian_category(urn:case:stacks:cat-a)",
    "additive_category(urn:case:stacks:cat-b)",
    "abelian_category(urn:case:stacks:cat-b)",
    "rf_everywhere_defined(urn:case:stacks:f)",
    "derived_functors_form_delta_functor(urn:case:stacks:f)",
    "negative_derived_functors_vanish(urn:case:stacks:f)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
    "proofGraph": {
      "nodes": [
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-1"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-1",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-10"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-10",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-11"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#enough_injectives"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-11",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-12"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#functor_between"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-12",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-13"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#homomorphism_on_hom_groups"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-13",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByHomGroups:7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-13"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByHomGroups",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-2"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-2",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-1",
            "urn:proof:assert:urn:mcp:case#fact-2"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByFiniteProducts",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-3"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_kernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-3",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-4"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_cokernels"
          },
          "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:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-5",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "definition": {
              "concept": "urn:stacks:clir:categories#abelian_category",
              "mode": "exact",
              "part": "sufficient"
            }
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#abelian_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662",
            "urn:proof:assert:urn:mcp:case#fact-3",
            "urn:proof:assert:urn:mcp:case#fact-4",
            "urn:proof:assert:urn:mcp:case#fact-5"
          ],
          "rule": "urn:stacks:clir:categories#abelian_category/sufficient",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-6"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-6",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-7"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-7",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-6",
            "urn:proof:assert:urn:mcp:case#fact-7"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByFiniteProducts",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-8"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_kernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-8",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-9"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_cokernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-9",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "definition": {
              "concept": "urn:stacks:clir:categories#abelian_category",
              "mode": "exact",
              "part": "sufficient"
            }
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#abelian_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d",
            "urn:proof:assert:urn:mcp:case#fact-10",
            "urn:proof:assert:urn:mcp:case#fact-8",
            "urn:proof:assert:urn:mcp:case#fact-9"
          ],
          "rule": "urn:stacks:clir:categories#abelian_category/sufficient",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:derived-functors#rf_everywhere_defined"
          },
          "evidence": [],
          "id": "urn:proof:apply:RFEverywhereDefined:f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:AdditiveByHomGroups:7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5",
            "urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a",
            "urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33",
            "urn:proof:assert:urn:mcp:case#fact-11",
            "urn:proof:assert:urn:mcp:case#fact-12"
          ],
          "rule": "urn:stacks:clir:derived-functors#RFEverywhereDefined",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            },
            "v1": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            },
            "v2": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:derived-functors#derived_functors_form_delta_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:DerivedFormDeltaFunctor:8f62e5cfbccfbc635c97b7ecd1e6ef2c33f53db9b99a642b5f0500ceabc5a9fd",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:RFEverywhereDefined:f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578"
          ],
          "rule": "urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:derived-functors#negative_derived_functors_vanish"
          },
          "evidence": [],
          "id": "urn:proof:apply:NegativeDerivedVanish:9302e91a8e09a730e8bd3f7715567533ed4485085d1d6b6b087034c3cb4d8ff2",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:RFEverywhereDefined:f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578"
          ],
          "rule": "urn:stacks:clir:derived-functors#NegativeDerivedVanish",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "constraint": "urn:stacks:clir:categories#abelian_category/necessary",
            "requirementStatus": "TRUE_ONLY",
            "status": "SATISFIED",
            "triggerStatus": "SATISFIED"
          },
          "evidence": [],
          "id": "urn:proof:constraint:abelian_category/necessary:75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484",
          "kind": "constraint_check",
          "premises": [
            "urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d",
            "urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33",
            "urn:proof:assert:urn:mcp:case#fact-10",
            "urn:proof:assert:urn:mcp:case#fact-8",
            "urn:proof:assert:urn:mcp:case#fact-9"
          ],
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
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SourcesExcerpts: 8

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
JSONRead only
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  "contentHash": "sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
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  "id": "urn:stacks:clir:derived-functors#ST_05SV",
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      "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
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{
  "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
JSONRead only
{
  "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}"
    }
  ]
}

tag/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_00ZY",
  "kind": "fragment",
  "locator": "tag/00ZY",
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    {
      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_0104",
  "kind": "fragment",
  "locator": "tag/0104",
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      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

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",
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      "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}"
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}

Packages in the snapshot

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
rf_everywhere_defined: TRUE_ONLY — установлено Выведено правом: additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f) Применены правила: AdditiveByFiniteProducts, AdditiveByHomGroups, abelian_category/sufficient, DerivedFormDeltaFunctor, NegativeDerivedVanish, RFEverywhereDefined Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 24 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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    "rf_everywhere_defined(urn:case:stacks:f)",
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    "programHash": "sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317",
    "resultHash": "sha256:2770eae7501dfd5877e5a407d53dd2a4ca8610caff6b6000c2eb16d65aecbba3",
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    "timezone": "Asia/Qyzylorda"
  },
  "rulesApplied": [
    "urn:stacks:clir:categories#AdditiveByFiniteProducts",
    "urn:stacks:clir:categories#AdditiveByHomGroups",
    "urn:stacks:clir:categories#abelian_category/sufficient",
    "urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor",
    "urn:stacks:clir:derived-functors#NegativeDerivedVanish",
    "urn:stacks:clir:derived-functors#RFEverywhereDefined"
  ],
  "signature": {
    "constants": {},
    "parameters": [
      {
        "labels": [],
        "name": "f",
        "type": {
          "name": "urn:stacks:clir:derived-functors#Functor"
        }
      }
    ],
    "predicate": "urn:stacks:clir:derived-functors#rf_everywhere_defined",
    "schemaVersion": "law.answers.signature/0.1",
    "types": {
      "urn:stacks:clir:category-theory#Functor": {
        "kind": "entity",
        "labels": [
          {
            "language": "en",
            "status": "official",
            "text": "functor"
          },
          {
            "language": "ru",
            "status": "unofficial",
            "text": "функтор"
          }
        ],
        "namespace": "urn:stacks:clir:category-theory",
        "package": "stacks.category_theory"
      },
      "urn:stacks:clir:derived-functors#Functor": {
        "kind": "alias",
        "labels": [],
        "namespace": "urn:stacks:clir:derived-functors",
        "package": "stacks.derived_functors",
        "target": "urn:stacks:clir:category-theory#Functor"
      }
    },
    "vocab": {}
  },
  "vulnerableTo": [],
  "whyNot": []
}

Execution · JSON

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Download JSON ↓

Display metadata

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Download JSON ↓

JSON · calculations, sources and exact data

JSONRead only
{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 16,
      "fragments": [
        "urn:stacks:clir:derived-functors#ST_05SV",
        "urn:stacks:clir:derived-functors#ST_05T4",
        "urn:stacks:clir:derived-functors#ST_05TD",
        "urn:stacks:clir:derived-functors#ST_05TE",
        "urn:stacks:clir:derived-functors#ST_05TI"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:derived-functors",
      "package": "stacks-derived-functors",
      "title": "Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"
    },
    {
      "contributed": true,
      "fragmentCount": 11,
      "fragments": [
        "urn:stacks:clir:categories#ST_00ZY",
        "urn:stacks:clir:categories#ST_0104",
        "urn:stacks:clir:categories#ST_0109"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:categories",
      "package": "stacks-categories",
      "title": "Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина"
    },
    {
      "contributed": false,
      "fragmentCount": 17,
      "fragments": [],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:category-theory",
      "package": "stacks-category-theory",
      "title": "Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина"
    }
  ],
  "caseHash": "sha256:7299d7d4001b7f6154f0372b741e9c7898ae828311ac9148a175f4d580767b11",
  "codeHash": "sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699",
  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317",
  "resultHash": "sha256:2770eae7501dfd5877e5a407d53dd2a4ca8610caff6b6000c2eb16d65aecbba3",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}
evaluation SHA-256
sha256:a3f3f46f1f498038c817d001dd0771ae51bca12d948d12d42120af63d67cc4be
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "homomorphism_on_hom_groups"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "rf_everywhere_defined",
  "proof": true
}

Три посылки: обе категории абелевы, достаточно инъективных объектов, функтор аддитивен.

Condition

Инъективная резольвента существует

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

the object has an injective resolution

x

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

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

    x: xc: cat-a

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:x"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:x",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "object_of"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "has_injective_resolution",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path11 steps

  1. 1

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-a

    case fact
  2. 2

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-a

    case fact
  3. 3

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-a

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  4. 4

    all kernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  5. 5

    all cokernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  6. 6

    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:cat-a

    case fact
  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:cat-a

    tag 0109

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

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: urn:case:stacks:cat-a

    case fact
  9. 9

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

    x: urn:case:stacks:x; c: urn:case:stacks:cat-a

    case fact
  10. 10

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

    the object has an injective resolution: a: urn:case:stacks:x

    tag 013K, ST 0138

    Identifier
    urn:stacks:clir:derived-functors#InjectiveResolutionsExist
    rule
  11. 11

    Query evaluation

    query

verified by the engine: 4 · case fact: 7 · Full graph: 12 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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 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

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

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

Derived result for this query

  • the object has an injective resolution

    a: x
Other derived facts2

  • 0104: the category is additive

    c: cat-a

  • 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: cat-a
0104: the category is additive
c
urn:case:stacks:cat-a
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:cat-a
the object has an injective resolution
a
x

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.

Proof graph

Proof graph · 5 layer
query_evaluationhas_injective_resolutionrule_applicationInjectiveResolutionsExistrule_applicationabelian_category/sufficientassertionenough_injectivesassertionobject_ofrule_applicationAdditiveByFiniteProductsassertionhas_all_kernelsassertionhas_all_cokernelsassertioncoimage_to_image_isomorphismassertionpreadditive_categoryassertionhas_finite_products

Proof nodes: 12 · assertion 7, rule_application 3, constraint_check 1, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
attributes
—
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-4
attributes
assertion
fact-4
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-4
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-5
attributes
assertion
fact-5
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-5
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
Type
rule_application
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • fact-3
  • fact-4
  • fact-5
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-6
attributes
assertion
fact-6
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
enough_injectives
evidence
—
Identifier
fact-6
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-7
attributes
assertion
fact-7
conclusion
Arguments
  • Identifier
    x
    Type
    entity_ref
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
object_of
evidence
—
Identifier
fact-7
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:InjectiveResolutionsExist:53d2feed021dbceb90a5871db741af6dcd18bc84406671769dd23eb2a521606e
attributes
—
conclusion
Arguments
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_injective_resolution
evidence
—
Identifier
53d2feed021dbceb90a5871db741af6dcd18bc84406671769dd23eb2a521606e
Type
rule_application
Premises
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • fact-6
  • fact-7
Rule
InjectiveResolutionsExist
sourceAnchors
—
substitution
v0
Identifier
x
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
Type
constraint_check
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • fact-3
  • fact-4
  • fact-5
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    x
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_injective_resolution
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • 53d2feed021dbceb90a5871db741af6dcd18bc84406671769dd23eb2a521606e
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "derived": [
    "additive_category(urn:case:stacks:cat-a)",
    "abelian_category(urn:case:stacks:cat-a)",
    "has_injective_resolution(urn:case:stacks:x)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
    "proofGraph": {
      "nodes": [
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-1"
          },
          "conclusion": {
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              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-1",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-2"
          },
          "conclusion": {
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              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-2",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
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SourcesExcerpts: 4

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
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  "contentHash": "sha256:9698ad04b03a0ca888a87e8f88e9925a856764046ae30cfb569730f33de5c92d",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:derived-functors#ST_013K",
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  "locator": "tag/013K",
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      "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/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
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{
  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:categories#ST_00ZY",
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    {
      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
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}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
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  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:categories#ST_0104",
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      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
    }
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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",
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      "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}"
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}

Packages in the snapshot

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
has_injective_resolution: TRUE_ONLY — установлено Выведено правом: additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); has_injective_resolution(urn:case:stacks:x) Применены правила: AdditiveByFiniteProducts, abelian_category/sufficient, InjectiveResolutionsExist Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 12 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

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          "text": "001S: products \\(x × y\\) exist for all pairs of objects, hence all finite products"
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      "kind": "symbol_decl",
      "labels": [
        {
          "language": "en",
          "status": "official",
          "text": "the object belongs to the category: \\(x ∈ Ob(C)\\)"
        },
        {
          "language": "ru",
          "status": "unofficial",
          "text": "объект принадлежит категории: \\(x ∈ Ob(C)\\)"
        }
      ],
      "name": "object_of",
      "package": "urn:stacks:clir:category-theory",
      "parameters": [
        {
          "id": "urn:stacks:clir:category-theory#object_of/arg/x",
          "labels": [],
          "name": "x",
          "type": {
            "name": "urn:stacks:clir:category-theory#Obj"
          }
        },
        {
          "id": "urn:stacks:clir:category-theory#object_of/arg/c",
          "labels": [],
          "name": "c",
          "type": {
            "name": "urn:stacks:clir:category-theory#Category"
          }
        }
      ],
      "symbolKind": "relation"
    },
    {
      "id": "urn:stacks:clir:derived-functors#InjectiveResolutionsExist",
      "kind": "rule",
      "labels": [
        {
          "language": "en",
          "status": "official",
          "text": "013K (1): in an abelian category with enough injectives any object has an injective resolution"
        },
        {
          "language": "ru",
          "status": "unofficial",
          "text": "013K (1): в абелевой категории с достаточным запасом инъективных у всякого объекта есть инъективная резольвента"
        }
      ],
      "package": "urn:stacks:clir:derived-functors",
      "strength": "strict"
    },
    {
      "id": "urn:stacks:clir:derived-functors#has_injective_resolution",
      "kind": "symbol_decl",
      "labels": [
        {
          "language": "en",
          "status": "official",
          "text": "the object has an injective resolution"
        },
        {
          "language": "ru",
          "status": "unofficial",
          "text": "у объекта есть инъективная резольвента"
        }
      ],
      "name": "has_injective_resolution",
      "package": "urn:stacks:clir:derived-functors",
      "parameters": [
        {
          "id": "urn:stacks:clir:derived-functors#has_injective_resolution/arg/a",
          "labels": [],
          "name": "a",
          "type": {
            "name": "urn:stacks:clir:derived-functors#Obj"
          }
        }
      ],
      "symbolKind": "relation"
    }
  ],
  "tables": []
}

JSON · calculations, sources and exact data

JSONRead only
{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 16,
      "fragments": [
        "urn:stacks:clir:derived-functors#ST_013K"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:derived-functors",
      "package": "stacks-derived-functors",
      "title": "Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"
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        "urn:stacks:clir:categories#ST_0104",
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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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      "title": "Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина"
    }
  ],
  "caseHash": "sha256:16d96832d4b46160686e000373db91605fb7fe8a63485616e294fea65e774545",
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  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317",
  "resultHash": "sha256:f65c6402e0f1650ff812e9ad868c34d07a4a8a444637c974636851f05dfd808d",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
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}
evaluation SHA-256
sha256:d7027c8e083c40d975e580372c09b698fd7fc16444312120dbf99a9be532189b
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:x"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:x",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "object_of"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "has_injective_resolution",
  "proof": true
}

Достаточность инъективных объектов даёт резольвенту для каждого объекта категории.

Condition

R¹ по инъективной резольвенте

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

015A: RnF(A)=Hn(RF(A[0]))R^nF(A) = H^n(RF(A[0])) is the given object

f1xh1

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-b

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-b

  • all kernels exist in the category

    c: cat-b

  • all cokernels exist in the category

    c: cat-b

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-b

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

  • the functor goes from the first category to the second: F:A→BF : A → B

    f: fa: cat-ab: cat-b

  • F:Mor(x,y)→Mor(F(x),F(y))F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all objects x,yx, y

    f: f

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

    x: xc: cat-a

  • K•K^• is a complex in the category

    k: ic: cat-a

  • 013I (1): In=0I^n = 0 for n<0n < 0

    k: i

  • 013I (2): each InI^n is an injective object of the category

    k: i

  • 013I (3): the map A→I0A → I^0 is an isomorphism onto Ker(d0)Ker(d^0)

    a: xk: i

  • 013I (4): Hi(I•)=0H^i(I^•) = 0 for i>0i > 0

    k: i

  • F(K•)F(K^•) is the given complex: FF applied termwise to K•K^•

    f: fk: ifk: f-i

  • Hn(K•)H^n(K^•) is the given object

    k: f-in: 1h: h1

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f",
    1,
    "urn:case:stacks:x",
    "urn:case:stacks:h1"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "homomorphism_on_hom_groups"
    },
    {
      "args": [
        "urn:case:stacks:x",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "object_of"
    },
    {
      "args": [
        "urn:case:stacks:i",
        "urn:case:stacks:cat-a"
      ],
      "predicate": "complex_in"
    },
    {
      "args": [
        "urn:case:stacks:i"
      ],
      "predicate": "vanishes_in_negative_degrees"
    },
    {
      "args": [
        "urn:case:stacks:i"
      ],
      "predicate": "terms_injective"
    },
    {
      "args": [
        "urn:case:stacks:x",
        "urn:case:stacks:i"
      ],
      "predicate": "augmentation_onto_kernel"
    },
    {
      "args": [
        "urn:case:stacks:i"
      ],
      "predicate": "acyclic_in_positive_degrees"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:i",
        "urn:case:stacks:f-i"
      ],
      "predicate": "applied_complex"
    },
    {
      "args": [
        "urn:case:stacks:f-i",
        1,
        "urn:case:stacks:h1"
      ],
      "predicate": "cohomology_of"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "right_derived",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path16 steps

  1. 1

    the functor goes from the first category to the second: F:A→BF : A → B

    f: urn:case:stacks:f; a: urn:case:stacks:cat-a; b: urn:case:stacks:cat-b

    case fact
  2. 2

    F:Mor(x,y)→Mor(F(x),F(y))F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all objects x,yx, y

    f: urn:case:stacks:f

    case fact
  3. 3

    00ZY: a functor of preadditive categories is additive if it is a homomorphism of abelian groups on morphism sets

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

    tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByHomGroups
    rule
  4. 4

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

    x: urn:case:stacks:x; c: urn:case:stacks:cat-a

    case fact
  5. 5

    K•K^• is a complex in the category

    k: urn:case:stacks:i; c: urn:case:stacks:cat-a

    case fact
  6. 6

    013I (1): In=0I^n = 0 for n<0n < 0

    k: urn:case:stacks:i

    case fact
  7. 7

    013I (2): each InI^n is an injective object of the category

    k: urn:case:stacks:i

    case fact
  8. 8

    013I (3): the map A→I0A → I^0 is an isomorphism onto Ker(d0)Ker(d^0)

    a: urn:case:stacks:x; k: urn:case:stacks:i

    case fact
  9. 9

    013I (4): Hi(I•)=0H^i(I^•) = 0 for i>0i > 0

    k: urn:case:stacks:i

    case fact
  10. 10

    013I: an injective resolution of AA is a complex I•I^• with A→I0A → I^0 such that (1) In=0I^n = 0 for n<0n < 0 , (2) each InI^n is injective, (3) A→I0A → I^0 is an isomorphism onto Ker(d0)Ker(d^0) , (4) Hi(I•)=0H^i(I^•) = 0 for i>0i > 0

    k: urn:case:stacks:i; a: urn:case:stacks:x

    tag 013I

    Identifier
    urn:stacks:clir:derived-functors#injective_resolution/sufficient
    rule
  11. 11

    05TH (1) with 013I: an injective resolution I•I^• of AA is a bounded below complex of injectives, hence computes RFRF for any additive F:A→BF : A → B

    05SX: the complex K•K^• computes RFRF : RFRF is defined at K•K^• and F(K•)→RF(K•)F(K^•) → RF(K^•) is an isomorphism: k: urn:case:stacks:i; f: urn:case:stacks:f

    tag 05TH, tag 013I

    Identifier
    urn:stacks:clir:derived-functors#InjectiveResolutionComputesRF
    rule
  12. 12

    013I: an injective resolution A→I•A → I^• is a quasi-isomorphism A[0]→I•A[0] → I^• , and I•I^• is bounded below by (1)

    K•K^• resolves AA : K•K^• is bounded below and A[0]→K•A[0] → K^• is a quasi-isomorphism: k: urn:case:stacks:i; a: urn:case:stacks:x

    tag 013I

    Identifier
    urn:stacks:clir:derived-functors#InjectiveResolutionResolves
    rule
  13. 13

    F(K•)F(K^•) is the given complex: FF applied termwise to K•K^•

    f: urn:case:stacks:f; k: urn:case:stacks:i; fk: urn:case:stacks:f-i

    case fact
  14. 14

    Hn(K•)H^n(K^•) is the given object

    k: urn:case:stacks:f-i; n: 1; h: urn:case:stacks:h1

    case fact
  15. 15

    015A with 05SX: if K•K^• resolves AA and computes RFRF , then RnF(A)=Hn(F(K•))R^nF(A) = H^n(F(K^•))

    015A: RnF(A)=Hn(RF(A[0]))R^nF(A) = H^n(RF(A[0])) is the given object: f: urn:case:stacks:f; n: 1; a: urn:case:stacks:x; h: urn:case:stacks:h1

    tag 015A, tag 05SX, tag 013I

    Identifier
    urn:stacks:clir:derived-functors#RightDerivedByResolution
    rule
  16. 16

    Query evaluation

    query

verified by the engine: 6 · case fact: 10 · Full graph: 38 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

  • 00ZY: a functor of preadditive categories is additive if it is a homomorphism of abelian groups on morphism sets

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

  • 05TH (1) with 013I: an injective resolution I•I^• of AA is a bounded below complex of injectives, hence computes RFRF for any additive F:A→BF : A → B

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

  • 013I: an injective resolution A→I•A → I^• is a quasi-isomorphism A[0]→I•A[0] → I^• , and I•I^• is bounded below by (1)

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

  • 015A with 05SX: if K•K^• resolves AA and computes RFRF , then RnF(A)=Hn(F(K•))R^nF(A) = H^n(F(K^•))

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

  • 013I: an injective resolution of AA is a complex I•I^• with A→I0A → I^0 such that (1) In=0I^n = 0 for n<0n < 0 , (2) each InI^n is injective, (3) A→I0A → I^0 is an isomorphism onto Ker(d0)Ker(d^0) , (4) Hi(I•)=0H^i(I^•) = 0 for i>0i > 0

    Identifier
    urn:stacks:clir:derived-functors#injective_resolution/sufficient
Other rules in the evaluation6

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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 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

  • 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

  • 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

  • 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

Derived result for this query

  • 015A: RnF(A)=Hn(RF(A[0]))R^nF(A) = H^n(RF(A[0])) is the given object

    f: fn: 1a: xh: h1
Other derived facts8

  • the functor is additive

    f: f

  • cochain complex

    k: i

    Subject shared by the facts below

  • 013I: an injective resolution of AA is a complex I•I^• with A→I0A → I^0 such that (1) In=0I^n = 0 for n<0n < 0 , (2) each InI^n is injective, (3) A→I0A → I^0 is an isomorphism onto Ker(d0)Ker(d^0) , (4) Hi(I•)=0H^i(I^•) = 0 for i>0i > 0

    a: x

  • 05SX: the complex K•K^• computes RFRF : RFRF is defined at K•K^• and F(K•)→RF(K•)F(K^•) → RF(K^•) is an isomorphism

    f: f

  • K•K^• resolves AA : K•K^• is bounded below and A[0]→K•A[0] → K^• is a quasi-isomorphism

    a: x

  • the object has an injective resolution

    a: x

  • 05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    f: f

  • 05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)

    f: f

  • 05TD (1): RiF=0R^iF = 0 for i<0i < 0

    f: f
the functor is additive
f
urn:case:stacks:f
013I: an injective resolution of AA is a complex I•I^• with A→I0A → I^0 such that (1) In=0I^n = 0 for n<0n < 0 , (2) each InI^n is injective, (3) A→I0A → I^0 is an isomorphism onto Ker(d0)Ker(d^0) , (4) Hi(I•)=0H^i(I^•) = 0 for i>0i > 0
ka
urn:case:stacks:ix
05SX: the complex K•K^• computes RFRF : RFRF is defined at K•K^• and F(K•)→RF(K•)F(K^•) → RF(K^•) is an isomorphism
kf
urn:case:stacks:if
K•K^• resolves AA : K•K^• is bounded below and A[0]→K•A[0] → K^• is a quasi-isomorphism
ka
urn:case:stacks:ix
Relationship graph
urn:case:stacks:furn:case:stacks:x
015A: RnF(A)=Hn(RF(A[0]))R^nF(A) = H^n(RF(A[0])) is the given object
fnah
f1xh1
the object has an injective resolution
a
x
05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined
f
f
05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)
f
f
05TD (1): RiF=0R^iF = 0 for i<0i < 0
f
f

4 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.

Proof graph

Proof graph · 5 layer
query_evaluationright_derivedrule_applicationRightDerivedByResolutionrule_applicationInjectiveResolutionComputesRFrule_applicationInjectiveResolutionResolvesassertionfunctor_betweenassertionobject_ofassertionapplied_complexassertioncohomology_ofrule_applicationAdditiveByHomGroupsrule_applicationinjective_resolution/sufficientassertioncomplex_inassertionhomomorphism_on_hom_groupsassertionvanishes_in_negative_degreesassertionterms_injectiveassertionaugmentation_onto_kernelassertionacyclic_in_positive_degrees

Proof nodes: 38 · assertion 21, rule_application 13, constraint_check 3, query_evaluation 1

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Proof reference
mcp
Original reasoning · JSON

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SourcesExcerpts: 13

tag/013I

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. Let A∈Ob(𝒜)A \in \Ob(\mathcal{A}) . An injective resolution of AA is a complex I•I^\bullet together with a map A→I0A \to I^0 such that:

  • We have In=0I^n = 0 for n<0n < 0 .

  • Each InI^n is an injective object of 𝒜\mathcal{A} .

  • The map A→I0A \to I^0 is an isomorphism onto Ker(d0)\Ker(d^0) .

  • We have Hi(I•)=0H^i(I^\bullet) = 0 for i>0i > 0 . Hence A[0]→I•A[0] \to I^\bullet is a quasi-isomorphism. In other words the complex

    …→0→A→I0→I1→…\ldots \to 0 \to A \to I^0 \to I^1 \to \ldots

    is acyclic. Let K•K^\bullet be a complex in 𝒜\mathcal{A} . An injective resolution of K•K^\bullet is a complex I•I^\bullet together with a map α:K•→I•\alpha : K^\bullet \to I^\bullet of complexes such that

  • We have In=0I^n = 0 for n≪0n \ll 0 , i.e., I•I^\bullet is bounded below.

  • Each InI^n is an injective object of 𝒜\mathcal{A} .

  • The map α:K•→I•\alpha : K^\bullet \to I^\bullet is a quasi-isomorphism.

Original data · JSON
JSONRead only
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  "contentHash": "sha256:d7c48bc941098921fd33924bd407abc5291d88b9236a485cdaaf7aa25e9d0832",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:derived-functors#ST_013I",
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      "contentHash": "sha256:7330949afd1629e09a53e8833569efe91f5d8ca6b7368e0a68a87d0caa530934",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-injective-resolution}\nLet $\\mathcal{A}$ be an abelian category.\nLet $A \\in \\Ob(\\mathcal{A})$.\nAn {\\it injective resolution of $A$} is a complex\n$I^\\bullet$ together with a map $A \\to I^0$ such\nthat:\n\\begin{enumerate}\n\\item We have $I^n = 0$ for $n < 0$.\n\\item Each $I^n$ is an injective object of $\\mathcal{A}$.\n\\item The map $A \\to I^0$ is an isomorphism onto $\\Ker(d^0)$.\n\\item We have $H^i(I^\\bullet) = 0$ for $i > 0$.\n\\end{enumerate}\nHence $A[0] \\to I^\\bullet$ is a quasi-isomorphism.\nIn other words the complex\n$$\n\\ldots \\to 0 \\to A \\to I^0 \\to I^1 \\to \\ldots\n$$\nis acyclic.\nLet $K^\\bullet$ be a complex in $\\mathcal{A}$.\nAn {\\it injective resolution of $K^\\bullet$} is a complex\n$I^\\bullet$ together with a map $\\alpha : K^\\bullet \\to I^\\bullet$\nof complexes such that\n\\begin{enumerate}\n\\item We have $I^n = 0$ for $n \\ll 0$, i.e., $I^\\bullet$ is bounded below.\n\\item Each $I^n$ is an injective object of $\\mathcal{A}$.\n\\item The map $\\alpha : K^\\bullet \\to I^\\bullet$ is a\nquasi-isomorphism.\n\\end{enumerate}\n\\end{definition}"
    }
  ]
}

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",
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  "locator": "tag/013K",
  "package": "urn:stacks:clir:derived-functors",
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    {
      "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/015A

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. Assume RF:D+(𝒜)→D+(ℬ)RF : D^{+}(\mathcal{A}) \to D^{+}(\mathcal{B}) is everywhere defined. Let i∈𝐙i \in \mathbf{Z} . The ii th right derived functor RiFR^iF of FF is the functor

RiF=Hi∘RF:𝒜⟶ℬR^iF = H^i \circ RF : \mathcal{A} \longrightarrow \mathcal{B}
Original data · JSON
JSONRead only
{
  "contentHash": "sha256:713c25b819cf11bf73664db0382e1e6093ee2fc2aef860be0a5c1844def10500",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:derived-functors#ST_015A",
  "kind": "fragment",
  "locator": "tag/015A",
  "package": "urn:stacks:clir:derived-functors",
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    {
      "contentHash": "sha256:7ecd153a29139bad7c85e08acd4818c3aedb736903c0a95c5400227dea7d508d",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-higher-derived-functors}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories. Assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $i \\in \\mathbf{Z}$.\nThe {\\it $i$th right derived functor $R^iF$ of $F$} is the functor\n$$\nR^iF = H^i \\circ RF :\n\\mathcal{A}\n\\longrightarrow\n\\mathcal{B}\n$$\n\\end{definition}"
    }
  ]
}

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
JSONRead only
{
  "contentHash": "sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
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  "id": "urn:stacks:clir:derived-functors#ST_05SV",
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  "locator": "tag/05SV",
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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/05SX

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

In Situation .

  • An object XX of 𝒟\mathcal{D} computes RFRF if RFRF is defined at XX and the canonical map F(X)→RF(X)F(X) \to RF(X) is an isomorphism.

  • An object XX of 𝒟\mathcal{D} computes LFLF if LFLF is defined at XX and the canonical map LF(X)→F(X)LF(X) \to F(X) is an isomorphism.

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:6d9640352974ec06f445076630ee0495e5463224a1f554aea25ab8abb8f231b0",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:derived-functors#ST_05SX",
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    {
      "contentHash": "sha256:a12a542d90fce40869d878a5e1f1be4adaed0407a78b0a5d0febeb73e30ca0b5",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-computes}\nIn\nSituation \\ref{situation-derived-functor}.\n\\begin{enumerate}\n\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $RF$ if $RF$ is defined\nat $X$ and the canonical map $F(X) \\to RF(X)$ is an isomorphism.\n\\item An object $X$ of $\\mathcal{D}$ {\\it computes} $LF$ if $LF$ is defined\nat $X$ and the canonical map $LF(X) \\to F(X)$ is an isomorphism.\n\\end{enumerate}\n\\end{definition}"
    }
  ]
}

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
JSONRead only
{
  "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",
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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",
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    {
      "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/05TH

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. Let I∈Ob(𝒜)I \in \Ob(\mathcal{A}) be an injective object. Let I•I^\bullet be a bounded below complex of injectives in 𝒜\mathcal{A} .

  • I•I^\bullet computes RFRF relative to Qis+(𝒜)\text{Qis}^{+}(\mathcal{A}) for any exact functor F:K+(𝒜)→𝒟F : K^{+}(\mathcal{A}) \to \mathcal{D} into any triangulated category 𝒟\mathcal{D} .

  • II is right acyclic for any additive functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} into any abelian category ℬ\mathcal{B} .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:67d5bab826525b8b40aa6d952b68667e49a96a051a298a45f2890a4bfb11950e",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:derived-functors#ST_05TH",
  "kind": "fragment",
  "locator": "tag/05TH",
  "package": "urn:stacks:clir:derived-functors",
  "texts": [
    {
      "contentHash": "sha256:e032a20e4fb0e18232eca1300982b8c867068569810bf8e4474e48073cab1465",
      "language": "en",
      "status": "official",
      "text": "\\begin{lemma}\n\\label{lemma-injective-acyclic}\nLet $\\mathcal{A}$ be an abelian category.\nLet $I \\in \\Ob(\\mathcal{A})$ be an injective object.\nLet $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$\nfor any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto any triangulated category $\\mathcal{D}$.\n\\item $I$ is right acyclic for any additive functor\n$F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"
    }
  ]
}

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}"
    }
  ]
}

tag/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_00ZY",
  "kind": "fragment",
  "locator": "tag/00ZY",
  "package": "urn:stacks:clir:categories",
  "texts": [
    {
      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_0104",
  "kind": "fragment",
  "locator": "tag/0104",
  "package": "urn:stacks:clir:categories",
  "texts": [
    {
      "contentHash": "sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

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"
}

Packages in the snapshot

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
right_derived: TRUE_ONLY — установлено Выведено правом: additive_functor(urn:case:stacks:f); injective_resolution(urn:case:stacks:i, urn:case:stacks:x); computes_rf(urn:case:stacks:i, urn:case:stacks:f); resolution_of(urn:case:stacks:i, urn:case:stacks:x); right_derived(urn:case:stacks:f, 1, urn:case:stacks:x, urn:case:stacks:h1); has_injective_resolution(urn:case:stacks:x); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f) …и ещё 4 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: AdditiveByFiniteProducts, AdditiveByHomGroups, abelian_category/sufficient, DerivedFormDeltaFunctor, InjectiveResolutionComputesRF, InjectiveResolutionResolves, InjectiveResolutionsExist, NegativeDerivedVanish, RFEverywhereDefined, RightDerivedByResolution, injective_resolution/sufficient Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 38 узлов — поле evaluation готово для law_explain

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{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 16,
      "fragments": [
        "urn:stacks:clir:derived-functors#ST_013I",
        "urn:stacks:clir:derived-functors#ST_013K",
        "urn:stacks:clir:derived-functors#ST_015A",
        "urn:stacks:clir:derived-functors#ST_05SV",
        "urn:stacks:clir:derived-functors#ST_05SX",
        "urn:stacks:clir:derived-functors#ST_05T4",
        "urn:stacks:clir:derived-functors#ST_05TD",
        "urn:stacks:clir:derived-functors#ST_05TE",
        "urn:stacks:clir:derived-functors#ST_05TH",
        "urn:stacks:clir:derived-functors#ST_05TI"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:derived-functors",
      "package": "stacks-derived-functors",
      "title": "Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"
    },
    {
      "contributed": true,
      "fragmentCount": 11,
      "fragments": [
        "urn:stacks:clir:categories#ST_00ZY",
        "urn:stacks:clir:categories#ST_0104",
        "urn:stacks:clir:categories#ST_0109"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:categories",
      "package": "stacks-categories",
      "title": "Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина"
    },
    {
      "contributed": false,
      "fragmentCount": 17,
      "fragments": [],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:category-theory",
      "package": "stacks-category-theory",
      "title": "Категория, функтор, изоморфизм, пределы, точные и сопряжённые функторы по главе «Categories» The Stacks Project: основание словаря когомологий — вне юрисдикции государства — доктрина"
    }
  ],
  "caseHash": "sha256:f9ff449733f859e1cbd1ccd89d5c53fed6bcf354e717972026f4805b743e759b",
  "codeHash": "sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699",
  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317",
  "resultHash": "sha256:9a35ac0bc226bc93d7a84a40c17a9ee6d2997c40b69b0e950c6ebc3437a14391",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}
evaluation SHA-256
sha256:da8cfaaf7600ffce90fe2000d08ab9ab35941d180ffe573d88ef188855508fb9
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f",
    1,
    "urn:case:stacks:x",
    "urn:case:stacks:h1"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "homomorphism_on_hom_groups"
    },
    {
      "args": [
        "urn:case:stacks:x",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "object_of"
    },
    {
      "args": [
        "urn:case:stacks:i",
        "urn:case:stacks:cat-a"
      ],
      "predicate": "complex_in"
    },
    {
      "args": [
        "urn:case:stacks:i"
      ],
      "predicate": "vanishes_in_negative_degrees"
    },
    {
      "args": [
        "urn:case:stacks:i"
      ],
      "predicate": "terms_injective"
    },
    {
      "args": [
        "urn:case:stacks:x",
        "urn:case:stacks:i"
      ],
      "predicate": "augmentation_onto_kernel"
    },
    {
      "args": [
        "urn:case:stacks:i"
      ],
      "predicate": "acyclic_in_positive_degrees"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:i",
        "urn:case:stacks:f-i"
      ],
      "predicate": "applied_complex"
    },
    {
      "args": [
        "urn:case:stacks:f-i",
        1,
        "urn:case:stacks:h1"
      ],
      "predicate": "cohomology_of"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "right_derived",
  "proof": true
}

Первый производный считается как когомология комплекса, полученного применением функтора к резольвенте.

Condition

Инъективный объект ацикличен

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

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

fj

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-b

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-b

  • all kernels exist in the category

    c: cat-b

  • all cokernels exist in the category

    c: cat-b

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-b

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

  • the functor goes from the first category to the second: F:A→BF : A → B

    f: fa: cat-ab: cat-b

  • 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: f

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

    x: jc: cat-a

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

    j: j

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f",
    "urn:case:stacks:j"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "preserves_left_exactness"
    },
    {
      "args": [
        "urn:case:stacks:j",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "object_of"
    },
    {
      "args": [
        "urn:case:stacks:j"
      ],
      "package": "stacks-categories",
      "predicate": "lifting_property"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "higher_derived_vanish",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path26 steps

  1. 1

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-a

    case fact
  2. 2

    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:cat-b

    case fact
  3. 3

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: urn:case:stacks:cat-a

    case fact
  4. 4

    the functor goes from the first category to the second: F:A→BF : A → B

    f: urn:case:stacks:f; a: urn:case:stacks:cat-a; b: urn:case:stacks:cat-b

    case fact
  5. 5

    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:f

    case fact
  6. 6

    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:f

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactBySES
    rule
  7. 7

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

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

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive
    rule
  8. 8

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

    x: urn:case:stacks:j; c: urn:case:stacks:cat-a

    case fact
  9. 9

    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:j

    case fact
  10. 10

    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:j

    tag 0135

    Identifier
    urn:stacks:clir:categories#InjectiveByLifting
    rule
  11. 11

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-a

    case fact
  12. 12

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-a

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  13. 13

    all kernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  14. 14

    all cokernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  15. 15

    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:cat-a

    case fact
  16. 16

    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:cat-a

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  17. 17

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-b

    case fact
  18. 18

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-b

    case fact
  19. 19

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-b

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  20. 20

    all kernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  21. 21

    all cokernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  22. 22

    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:cat-b

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  23. 23

    05TH (2): an injective object II is right acyclic for any additive functor F:A→BF : A → B into an abelian category

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

    tag 05TH, tag 05T4

    Identifier
    urn:stacks:clir:derived-functors#InjectiveIsRightAcyclic
    rule
  24. 24

    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:f

    tag 05TI, tag 05SV, tag 05T4

    Identifier
    urn:stacks:clir:derived-functors#RFEverywhereDefined
    rule
  25. 25

    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:f; a: urn:case:stacks:j

    tag 015C

    Identifier
    urn:stacks:clir:derived-functors#HigherDerivedVanishForAcyclic
    rule
  26. 26

    Query evaluation

    query

verified by the engine: 11 · case fact: 15 · Full graph: 33 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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 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

  • 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

  • 05TH (2): an injective object II is right acyclic for any additive functor F:A→BF : A → B into an abelian category

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

  • 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
Other rules in the evaluation5

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

Derived result for this query

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

    f: fa: j
Other derived facts10

  • 0034 (1): the functor is left exact

    f: f

  • the functor is additive

    f: f

  • the object is injective

    j: j

  • the object has an injective resolution

    a: j

  • 015B (4): the sequence (RiF,δ)(R^iF, δ) is a universal δδ -functor (010S) from AA to BB

    f: f

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

    a: jf: f

  • 05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    f: f

  • 05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)

    f: f

  • 05TD (1): RiF=0R^iF = 0 for i<0i < 0

    f: f

  • 05TD (3): the map F→R0FF → R^0F is an isomorphism

    f: f
0034 (1): the functor is left exact
f
f
the functor is additive
f
urn:case:stacks:f
the object is injective
j
urn:case:stacks:j
the object has an injective resolution
a
j
015B (4): the sequence (RiF,δ)(R^iF, δ) is a universal δδ -functor (010S) from AA to BB
f
f
Relationship graph
urn:case:stacks:jurn:case:stacks:f
0157 (3): the object AA is right acyclic for FF : A[0]A[0] computes RFRF
af
jf
05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined
f
f
05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)
f
f
Relationship graph
urn:case:stacks:furn:case:stacks:j
015C: RiF(A)=0R^iF(A) = 0 for all i>0i > 0
fa
fj
05TD (1): RiF=0R^iF = 0 for i<0i < 0
f
f
05TD (3): the map F→R0FF → R^0F is an isomorphism
f
f

4 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.

Proof graph

Proof graph · 6 layer
query_evaluationhigher_derived_vanishrule_applicationHigherDerivedVanishForAcyclicrule_applicationInjectiveIsRightAcyclicrule_applicationLeftExactBySESrule_applicationRFEverywhereDefinedrule_applicationInjectiveByLiftingrule_applicationLeftExactIsAdditiverule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientassertionfunctor_betweenassertionobject_ofassertionpreserves_left_exactnessassertionenough_injectivesassertionlifting_propertyrule_applicationAdditiveByFiniteProductsassertionhas_all_kernelsassertionhas_all_cokernelsassertioncoimage_to_image_isomorphismrule_applicationAdditiveByFiniteProductsassertioncoimage_to_image_isomorphismassertionhas_all_kernelsassertionhas_all_cokernelsassertionpreadditive_categoryassertionhas_finite_productsassertionpreadditive_categoryassertionhas_finite_products

Proof nodes: 33 · assertion 15, rule_application 15, constraint_check 2, query_evaluation 1

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Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON

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SourcesExcerpts: 14

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
JSONRead only
{
  "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
JSONRead only
{
  "contentHash": "sha256:e250263a8b2f3813392cdbde3901795ceda78f89b9865dfdc2a7794d781e519a",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:derived-functors#ST_015C",
  "kind": "fragment",
  "locator": "tag/015C",
  "package": "urn:stacks:clir:derived-functors",
  "texts": [
    {
      "contentHash": "sha256:d0947fabef80aaa804f6f4635edca13dba8e17d1cdf28aacd25709217ffafa46",
      "language": "en",
      "status": "official",
      "text": "\\begin{lemma}\n\\label{lemma-F-acyclic}\nLet $F : \\mathcal{A} \\to \\mathcal{B}$ be an additive functor\nbetween abelian categories and assume\n$RF : D^{+}(\\mathcal{A}) \\to D^{+}(\\mathcal{B})$ is everywhere\ndefined. Let $A$ be an object of $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $A$ is right acyclic for $F$ if and only if\n$F(A) \\to R^0F(A)$ is an isomorphism and $R^iF(A) = 0$ for all $i > 0$,\n\\item if $F$ is left exact, then $A$ is right acyclic for $F$\nif and only if $R^iF(A) = 0$ for all $i > 0$.\n\\end{enumerate}\n\\end{lemma}"
    }
  ]
}

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
JSONRead only
{
  "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
JSONRead only
{
  "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/05TH

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. Let I∈Ob(𝒜)I \in \Ob(\mathcal{A}) be an injective object. Let I•I^\bullet be a bounded below complex of injectives in 𝒜\mathcal{A} .

  • I•I^\bullet computes RFRF relative to Qis+(𝒜)\text{Qis}^{+}(\mathcal{A}) for any exact functor F:K+(𝒜)→𝒟F : K^{+}(\mathcal{A}) \to \mathcal{D} into any triangulated category 𝒟\mathcal{D} .

  • II is right acyclic for any additive functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} into any abelian category ℬ\mathcal{B} .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:67d5bab826525b8b40aa6d952b68667e49a96a051a298a45f2890a4bfb11950e",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:derived-functors#ST_05TH",
  "kind": "fragment",
  "locator": "tag/05TH",
  "package": "urn:stacks:clir:derived-functors",
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    {
      "contentHash": "sha256:e032a20e4fb0e18232eca1300982b8c867068569810bf8e4474e48073cab1465",
      "language": "en",
      "status": "official",
      "text": "\\begin{lemma}\n\\label{lemma-injective-acyclic}\nLet $\\mathcal{A}$ be an abelian category.\nLet $I \\in \\Ob(\\mathcal{A})$ be an injective object.\nLet $I^\\bullet$ be a bounded below complex of injectives in $\\mathcal{A}$.\n\\begin{enumerate}\n\\item $I^\\bullet$ computes $RF$ relative to $\\text{Qis}^{+}(\\mathcal{A})$\nfor any exact functor $F : K^{+}(\\mathcal{A}) \\to \\mathcal{D}$\ninto any triangulated category $\\mathcal{D}$.\n\\item $I$ is right acyclic for any additive functor\n$F : \\mathcal{A} \\to \\mathcal{B}$ into any abelian category $\\mathcal{B}$.\n\\end{enumerate}\n\\end{lemma}"
    }
  ]
}

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}"
    }
  ]
}

tag/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_00ZY",
  "kind": "fragment",
  "locator": "tag/00ZY",
  "package": "urn:stacks:clir:categories",
  "texts": [
    {
      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_0104",
  "kind": "fragment",
  "locator": "tag/0104",
  "package": "urn:stacks:clir:categories",
  "texts": [
    {
      "contentHash": "sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

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

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
higher_derived_vanish: TRUE_ONLY — установлено Выведено правом: left_exact_functor(urn:case:stacks:f); additive_functor(urn:case:stacks:f); injective_object(urn:case:stacks:j); has_injective_resolution(urn:case:stacks:j); derived_functors_universal(urn:case:stacks:f); right_acyclic_for(urn:case:stacks:j, urn:case:stacks:f); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); higher_derived_vanish(urn:case:stacks:f, urn:case:stacks:j); negative_derived_functors_vanish(urn:case:stacks:f); r0_agrees_with_f(urn:case:stacks:f) …и ещё 4 выведенных фактов вне предмета вопроса (полный вывод — law_explain) Применены правила: AdditiveByFiniteProducts, InjectiveByLifting, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, HigherDerivedVanishForAcyclic, InjectiveIsRightAcyclic, InjectiveResolutionsExist, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 33 узлов — поле evaluation готово для law_explain

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        "urn:stacks:clir:derived-functors#ST_05SV",
        "urn:stacks:clir:derived-functors#ST_05T4",
        "urn:stacks:clir:derived-functors#ST_05TD",
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      "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"
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        "urn:stacks:clir:categories#ST_0135"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:categories",
      "package": "stacks-categories",
      "title": "Абелевы категории, точные функторы и инъективные объекты по главе «Homological Algebra» The Stacks Project: нижний слой словаря когомологий — вне юрисдикции государства — доктрина"
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      "namespace": "urn:stacks:clir:category-theory",
      "package": "stacks-category-theory",
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  "legalTime": "2026-09-06",
  "mode": "audit",
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}
evaluation SHA-256
sha256:6e0eddcf0aed8ec14fa826816d484cb45735200fe99db7335f88eca9a74e8552
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f",
    "urn:case:stacks:j"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "preserves_left_exactness"
    },
    {
      "args": [
        "urn:case:stacks:j",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "object_of"
    },
    {
      "args": [
        "urn:case:stacks:j"
      ],
      "package": "stacks-categories",
      "predicate": "lifting_property"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "higher_derived_vanish",
  "proof": true
}

На инъективном объекте производные положительной степени обращаются в нуль.

Condition

R⁰F совпадает с F при левой точности

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

05TD (3): the map F→R0FF → R^0F is an isomorphism

f

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-b

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-b

  • all kernels exist in the category

    c: cat-b

  • all cokernels exist in the category

    c: cat-b

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-b

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

  • the functor goes from the first category to the second: F:A→BF : A → B

    f: fa: cat-ab: cat-b

  • 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: f

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "preserves_left_exactness"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "r0_agrees_with_f",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path22 steps

  1. 1

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-a

    case fact
  2. 2

    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:cat-b

    case fact
  3. 3

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: urn:case:stacks:cat-a

    case fact
  4. 4

    the functor goes from the first category to the second: F:A→BF : A → B

    f: urn:case:stacks:f; a: urn:case:stacks:cat-a; b: urn:case:stacks:cat-b

    case fact
  5. 5

    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:f

    case fact
  6. 6

    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:f

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactBySES
    rule
  7. 7

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

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

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive
    rule
  8. 8

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-a

    case fact
  9. 9

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-a

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  10. 10

    all kernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  11. 11

    all cokernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  12. 12

    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:cat-a

    case fact
  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:cat-a

    tag 0109

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

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-b

    case fact
  15. 15

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-b

    case fact
  16. 16

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-b

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  17. 17

    all kernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  18. 18

    all cokernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  19. 19

    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:cat-b

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  20. 20

    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:f

    tag 05TI, tag 05SV, tag 05T4

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

    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:f

    tag 05TD

    Identifier
    urn:stacks:clir:derived-functors#R0AgreesIfLeftExact
    rule
  22. 22

    Query evaluation

    query

verified by the engine: 9 · case fact: 13 · Full graph: 27 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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 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
Other rules in the evaluation3

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

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

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

Derived result for this query

  • 05TD (3): the map F→R0FF → R^0F is an isomorphism

    f: f
Other derived facts10

  • 0034 (1): the functor is left exact

    f: f

  • the functor is additive

    f: f

  • 0104: the category is additive

    c: cat-a

  • 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: cat-a

  • 0104: the category is additive

    c: cat-b

  • 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: cat-b

  • 015B (4): the sequence (RiF,δ)(R^iF, δ) is a universal δδ -functor (010S) from AA to BB

    f: f

  • 05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    f: f

  • 05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)

    f: f

  • 05TD (1): RiF=0R^iF = 0 for i<0i < 0

    f: f
0034 (1): the functor is left exact
f
f
the functor is additive
f
urn:case:stacks:f
0104: the category is additive
c
urn:case:stacks:cat-a
urn:case:stacks:cat-b
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:cat-a
urn:case:stacks:cat-b
015B (4): the sequence (RiF,δ)(R^iF, δ) is a universal δδ -functor (010S) from AA to BB
f
f
05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined
f
f
05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)
f
f
05TD (1): RiF=0R^iF = 0 for i<0i < 0
f
f
05TD (3): the map F→R0FF → R^0F is an isomorphism
f
f

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.

Proof graph

Proof graph · 6 layer
query_evaluationr0_agrees_with_frule_applicationR0AgreesIfLeftExactrule_applicationLeftExactBySESrule_applicationRFEverywhereDefinedassertionpreserves_left_exactnessrule_applicationLeftExactIsAdditiverule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientassertionenough_injectivesassertionfunctor_betweenrule_applicationAdditiveByFiniteProductsassertionhas_all_kernelsassertionhas_all_cokernelsassertioncoimage_to_image_isomorphismrule_applicationAdditiveByFiniteProductsassertioncoimage_to_image_isomorphismassertionhas_all_kernelsassertionhas_all_cokernelsassertionpreadditive_categoryassertionhas_finite_productsassertionpreadditive_categoryassertionhas_finite_products

Proof nodes: 27 · assertion 13, rule_application 11, constraint_check 2, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-10
attributes
assertion
fact-10
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-10
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-11
attributes
assertion
fact-11
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
enough_injectives
evidence
—
Identifier
fact-11
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-12
attributes
assertion
fact-12
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    cat-a
    Type
    entity_ref
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
functor_between
evidence
—
Identifier
fact-12
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-13
attributes
assertion
fact-13
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preserves_left_exactness
evidence
—
Identifier
fact-13
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:LeftExactBySES:763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
left_exact_functor
evidence
—
Identifier
763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
Type
rule_application
Premises
  • fact-13
Rule
LeftExactBySES
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:LeftExactIsAdditive:4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_functor
evidence
—
Identifier
4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
Rule
LeftExactIsAdditive
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
attributes
—
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-4
attributes
assertion
fact-4
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-4
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-5
attributes
assertion
fact-5
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-5
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
Type
rule_application
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • fact-3
  • fact-4
  • fact-5
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-6
attributes
assertion
fact-6
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-6
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-7
attributes
assertion
fact-7
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-7
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
attributes
—
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
Type
rule_application
Premises
  • fact-6
  • fact-7
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-8
attributes
assertion
fact-8
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-8
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-9
attributes
assertion
fact-9
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-9
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
Type
rule_application
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • fact-10
  • fact-8
  • fact-9
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:DerivedUniversalDeltaFunctor:d4dce774b8eebd84a8bbb1a4222674523e7f4d3bdc1a48ce605711cb14709f7e
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
derived_functors_universal
evidence
—
Identifier
d4dce774b8eebd84a8bbb1a4222674523e7f4d3bdc1a48ce605711cb14709f7e
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
Rule
DerivedUniversalDeltaFunctor
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:RFEverywhereDefined:c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
rf_everywhere_defined
evidence
—
Identifier
c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Type
rule_application
Premises
  • 4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
Rule
RFEverywhereDefined
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:DerivedFormDeltaFunctor:4897774f16e9f0a93ddf301cff8ae4daf84d7c5fbc09f1dc30947c6a8ecff70d
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
derived_functors_form_delta_functor
evidence
—
Identifier
4897774f16e9f0a93ddf301cff8ae4daf84d7c5fbc09f1dc30947c6a8ecff70d
Type
rule_application
Premises
  • c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Rule
DerivedFormDeltaFunctor
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:NegativeDerivedVanish:cba76e4e91360036b33877f83bc4be3f3b0ec09c661fda986f45f17b90b6c383
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
negative_derived_functors_vanish
evidence
—
Identifier
cba76e4e91360036b33877f83bc4be3f3b0ec09c661fda986f45f17b90b6c383
Type
rule_application
Premises
  • c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Rule
NegativeDerivedVanish
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:R0AgreesIfLeftExact:94a87016946b5ee2cc8dd617e9553bc6b65015ed78aaaba4bb68185c0632a266
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
r0_agrees_with_f
evidence
—
Identifier
94a87016946b5ee2cc8dd617e9553bc6b65015ed78aaaba4bb68185c0632a266
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
  • c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Rule
R0AgreesIfLeftExact
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
Type
constraint_check
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-10
  • fact-8
  • fact-9
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
Type
constraint_check
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • fact-3
  • fact-4
  • fact-5
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
r0_agrees_with_f
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • 94a87016946b5ee2cc8dd617e9553bc6b65015ed78aaaba4bb68185c0632a266
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "derived": [
    "left_exact_functor(urn:case:stacks:f)",
    "additive_functor(urn:case:stacks:f)",
    "additive_category(urn:case:stacks:cat-a)",
    "abelian_category(urn:case:stacks:cat-a)",
    "additive_category(urn:case:stacks:cat-b)",
    "abelian_category(urn:case:stacks:cat-b)",
    "derived_functors_universal(urn:case:stacks:f)",
    "rf_everywhere_defined(urn:case:stacks:f)",
    "derived_functors_form_delta_functor(urn:case:stacks:f)",
    "negative_derived_functors_vanish(urn:case:stacks:f)",
    "r0_agrees_with_f(urn:case:stacks:f)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
    "proofGraph": {
      "nodes": [
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-1"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-1",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-10"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-10",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-11"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#enough_injectives"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-11",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-12"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#functor_between"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-12",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-13"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preserves_left_exactness"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-13",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#left_exact_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:LeftExactBySES:763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-13"
          ],
          "rule": "urn:stacks:clir:categories#LeftExactBySES",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:LeftExactIsAdditive:4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:LeftExactBySES:763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41"
          ],
          "rule": "urn:stacks:clir:categories#LeftExactIsAdditive",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-2"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-2",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-1",
            "urn:proof:assert:urn:mcp:case#fact-2"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByFiniteProducts",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-3"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_kernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-3",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-4"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_cokernels"
          },
          "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:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-5",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "definition": {
              "concept": "urn:stacks:clir:categories#abelian_category",
              "mode": "exact",
              "part": "sufficient"
            }
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#abelian_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a",
          "kind": "rule_application",
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SourcesExcerpts: 10

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} .

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  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:derived-functors#ST_015B",
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      "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}"
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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} .

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  "contentHash": "sha256:52324507289c1f24de386efd6c27f75fc4ed9e445c1c254d3de5f6f78d4e02be",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
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      "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}"
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}

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 .

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  "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
JSONRead only
{
  "contentHash": "sha256:174e81d1e6a2a741a7ca892fd32c0730c7a3e2200102395c172408bdb4b5453b",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
  "fragmentKind": "lemma",
  "id": "urn:stacks:clir:derived-functors#ST_05TD",
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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
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  "contentHash": "sha256:40b9cb4e0f5b604cc2daaa8085d1142dae0ac30e4bcc9adec87883443ec4e5d6",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
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  "id": "urn:stacks:clir:derived-functors#ST_05TE",
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  "locator": "tag/05TE",
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    {
      "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",
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  "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}"
    }
  ]
}

tag/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_00ZY",
  "kind": "fragment",
  "locator": "tag/00ZY",
  "package": "urn:stacks:clir:categories",
  "texts": [
    {
      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:categories#ST_0104",
  "kind": "fragment",
  "locator": "tag/0104",
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      "contentHash": "sha256:675cbb2c05ee57eb18db573e06c5dd58bc69ba38f1dc3b21733b091d62655a30",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

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"
}

Packages in the snapshot

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
r0_agrees_with_f: TRUE_ONLY — установлено Выведено правом: left_exact_functor(urn:case:stacks:f); additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); derived_functors_universal(urn:case:stacks:f); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f); r0_agrees_with_f(urn:case:stacks:f) Применены правила: AdditiveByFiniteProducts, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 27 узлов — поле evaluation готово для law_explain

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JSON · calculations, sources and exact data

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        "urn:stacks:clir:derived-functors#ST_05TD",
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      "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"
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      "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:c21563e6d8c2153d4e7b33a3cdb5e524b00231b05e64392424f9fc73509ce57b
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      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "preserves_left_exactness"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "r0_agrees_with_f",
  "proof": true
}

Совпадение нулевой степени с самим функтором — ровно свойство левой точности.

Condition

Длинная точная последовательность

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

015B (1): the short exact sequence 0→A→B→C→00 → A → B → C → 0 has an associated long exact sequence …→RiF(A)→RiF(B)→RiF(C)→Ri+1F(A)→…… → R^iF(A) → R^iF(B) → R^iF(C) → R^{i+1}F(A) → …

fs

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-b

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-b

  • all kernels exist in the category

    c: cat-b

  • all cokernels exist in the category

    c: cat-b

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-b

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

  • the functor goes from the first category to the second: F:A→BF : A → B

    f: fa: cat-ab: cat-b

  • 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: f

  • the sequence 0→A→B→C→00 → A → B → C → 0 is a short exact sequence in the category

    s: sc: cat-a

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f",
    "urn:case:stacks:s"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "preserves_left_exactness"
    },
    {
      "args": [
        "urn:case:stacks:s",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "short_exact_sequence"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "long_exact_sequence_of_derived",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path21 steps

  1. 1

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-a

    case fact
  2. 2

    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:cat-b

    case fact
  3. 3

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: urn:case:stacks:cat-a

    case fact
  4. 4

    the functor goes from the first category to the second: F:A→BF : A → B

    f: urn:case:stacks:f; a: urn:case:stacks:cat-a; b: urn:case:stacks:cat-b

    case fact
  5. 5

    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:f

    case fact
  6. 6

    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:f

    tag 010N

    Identifier
    urn:stacks:clir:categories#LeftExactBySES
    rule
  7. 7

    the sequence 0→A→B→C→00 → A → B → C → 0 is a short exact sequence in the category

    s: urn:case:stacks:s; c: urn:case:stacks:cat-a

    case fact
  8. 8

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-a

    case fact
  9. 9

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-a

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  10. 10

    all kernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  11. 11

    all cokernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  12. 12

    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:cat-a

    case fact
  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:cat-a

    tag 0109

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

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-b

    case fact
  15. 15

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-b

    case fact
  16. 16

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-b

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  17. 17

    all kernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  18. 18

    all cokernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  19. 19

    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:cat-b

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  20. 20

    015B (1): if AA is abelian with enough injectives and F:A→BF : A → B is left exact, every short exact sequence in AA has an associated long exact sequence of the RiFR^iF

    015B (1): the short exact sequence 0→A→B→C→00 → A → B → C → 0 has an associated long exact sequence …→RiF(A)→RiF(B)→RiF(C)→Ri+1F(A)→…… → R^iF(A) → R^iF(B) → R^iF(C) → R^{i+1}F(A) → …: f: urn:case:stacks:f; s: urn:case:stacks:s

    tag 015B, ST 010Q

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

    Query evaluation

    query

verified by the engine: 7 · case fact: 14 · Full graph: 29 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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 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

  • 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

  • 015B (1): if AA is abelian with enough injectives and F:A→BF : A → B is left exact, every short exact sequence in AA has an associated long exact sequence of the RiFR^iF

    Identifier
    urn:stacks:clir:derived-functors#LongExactSequence
Other rules in the evaluation6

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

    Identifier
    urn:stacks:clir:categories#LeftExactIsAdditive
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

  • 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

  • 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

Derived result for this query

  • 015B (1): the short exact sequence 0→A→B→C→00 → A → B → C → 0 has an associated long exact sequence …→RiF(A)→RiF(B)→RiF(C)→Ri+1F(A)→…… → R^iF(A) → R^iF(B) → R^iF(C) → R^{i+1}F(A) → …

    f: fs: s
Other derived facts11

  • 0034 (1): the functor is left exact

    f: f

  • the functor is additive

    f: f

  • 0104: the category is additive

    c: cat-a

  • 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: cat-a

  • 0104: the category is additive

    c: cat-b

  • 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: cat-b

  • 015B (4): the sequence (RiF,δ)(R^iF, δ) is a universal δδ -functor (010S) from AA to BB

    f: f

  • 05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    f: f

  • 05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)

    f: f

  • 05TD (1): RiF=0R^iF = 0 for i<0i < 0

    f: f

  • 05TD (3): the map F→R0FF → R^0F is an isomorphism

    f: f
0034 (1): the functor is left exact
f
f
the functor is additive
f
urn:case:stacks:f
0104: the category is additive
c
urn:case:stacks:cat-a
urn:case:stacks:cat-b
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:cat-a
urn:case:stacks:cat-b
015B (4): the sequence (RiF,δ)(R^iF, δ) is a universal δδ -functor (010S) from AA to BB
f
f
Relationship graph
urn:case:stacks:furn:case:stacks:s
015B (1): the short exact sequence 0→A→B→C→00 → A → B → C → 0 has an associated long exact sequence …→RiF(A)→RiF(B)→RiF(C)→Ri+1F(A)→…… → R^iF(A) → R^iF(B) → R^iF(C) → R^{i+1}F(A) → …
fs
fs
05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined
f
f
05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)
f
f
05TD (1): RiF=0R^iF = 0 for i<0i < 0
f
f
05TD (3): the map F→R0FF → R^0F is an isomorphism
f
f

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.

Proof graph

Proof graph · 5 layer
query_evaluationlong_exact_sequence_of_derivedrule_applicationLongExactSequencerule_applicationLeftExactBySESrule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientassertionenough_injectivesassertionfunctor_betweenassertionshort_exact_sequenceassertionpreserves_left_exactnessrule_applicationAdditiveByFiniteProductsassertionhas_all_kernelsassertionhas_all_cokernelsassertioncoimage_to_image_isomorphismrule_applicationAdditiveByFiniteProductsassertioncoimage_to_image_isomorphismassertionhas_all_kernelsassertionhas_all_cokernelsassertionpreadditive_categoryassertionhas_finite_productsassertionpreadditive_categoryassertionhas_finite_products

Proof nodes: 29 · assertion 14, rule_application 12, constraint_check 2, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-10
attributes
assertion
fact-10
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-10
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-11
attributes
assertion
fact-11
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
enough_injectives
evidence
—
Identifier
fact-11
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-12
attributes
assertion
fact-12
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    cat-a
    Type
    entity_ref
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
functor_between
evidence
—
Identifier
fact-12
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-13
attributes
assertion
fact-13
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preserves_left_exactness
evidence
—
Identifier
fact-13
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:LeftExactBySES:763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
left_exact_functor
evidence
—
Identifier
763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
Type
rule_application
Premises
  • fact-13
Rule
LeftExactBySES
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:LeftExactIsAdditive:4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_functor
evidence
—
Identifier
4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
Rule
LeftExactIsAdditive
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-14
attributes
assertion
fact-14
conclusion
Arguments
  • Identifier
    s
    Type
    entity_ref
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
short_exact_sequence
evidence
—
Identifier
fact-14
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
attributes
—
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-4
attributes
assertion
fact-4
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-4
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-5
attributes
assertion
fact-5
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-5
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
Type
rule_application
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • fact-3
  • fact-4
  • fact-5
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-6
attributes
assertion
fact-6
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-6
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-7
attributes
assertion
fact-7
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-7
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
attributes
—
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
Type
rule_application
Premises
  • fact-6
  • fact-7
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-8
attributes
assertion
fact-8
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-8
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-9
attributes
assertion
fact-9
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-9
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
Type
rule_application
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • fact-10
  • fact-8
  • fact-9
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:DerivedUniversalDeltaFunctor:d4dce774b8eebd84a8bbb1a4222674523e7f4d3bdc1a48ce605711cb14709f7e
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
derived_functors_universal
evidence
—
Identifier
d4dce774b8eebd84a8bbb1a4222674523e7f4d3bdc1a48ce605711cb14709f7e
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
Rule
DerivedUniversalDeltaFunctor
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:LongExactSequence:efb175f6b16e7106615dceec1f183d2346ba85a371691cc973750a224da696c0
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    s
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
long_exact_sequence_of_derived
evidence
—
Identifier
efb175f6b16e7106615dceec1f183d2346ba85a371691cc973750a224da696c0
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
  • fact-14
Rule
LongExactSequence
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
v3
Identifier
s
Type
entity_ref
rule_application · urn:proof:apply:RFEverywhereDefined:c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
rf_everywhere_defined
evidence
—
Identifier
c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Type
rule_application
Premises
  • 4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
Rule
RFEverywhereDefined
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:DerivedFormDeltaFunctor:4897774f16e9f0a93ddf301cff8ae4daf84d7c5fbc09f1dc30947c6a8ecff70d
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
derived_functors_form_delta_functor
evidence
—
Identifier
4897774f16e9f0a93ddf301cff8ae4daf84d7c5fbc09f1dc30947c6a8ecff70d
Type
rule_application
Premises
  • c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Rule
DerivedFormDeltaFunctor
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:NegativeDerivedVanish:cba76e4e91360036b33877f83bc4be3f3b0ec09c661fda986f45f17b90b6c383
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
negative_derived_functors_vanish
evidence
—
Identifier
cba76e4e91360036b33877f83bc4be3f3b0ec09c661fda986f45f17b90b6c383
Type
rule_application
Premises
  • c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Rule
NegativeDerivedVanish
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:R0AgreesIfLeftExact:94a87016946b5ee2cc8dd617e9553bc6b65015ed78aaaba4bb68185c0632a266
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
r0_agrees_with_f
evidence
—
Identifier
94a87016946b5ee2cc8dd617e9553bc6b65015ed78aaaba4bb68185c0632a266
Type
rule_application
Premises
  • 763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41
  • c71991058587b13c0e4aafcf8fb5722cc3b3c1ec8d591ff26990432d6988e3ff
Rule
R0AgreesIfLeftExact
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
Type
constraint_check
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-10
  • fact-8
  • fact-9
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
Type
constraint_check
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • fact-3
  • fact-4
  • fact-5
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    s
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
long_exact_sequence_of_derived
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • efb175f6b16e7106615dceec1f183d2346ba85a371691cc973750a224da696c0
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "derived": [
    "left_exact_functor(urn:case:stacks:f)",
    "additive_functor(urn:case:stacks:f)",
    "additive_category(urn:case:stacks:cat-a)",
    "abelian_category(urn:case:stacks:cat-a)",
    "additive_category(urn:case:stacks:cat-b)",
    "abelian_category(urn:case:stacks:cat-b)",
    "derived_functors_universal(urn:case:stacks:f)",
    "long_exact_sequence_of_derived(urn:case:stacks:f, urn:case:stacks:s)",
    "rf_everywhere_defined(urn:case:stacks:f)",
    "derived_functors_form_delta_functor(urn:case:stacks:f)",
    "negative_derived_functors_vanish(urn:case:stacks:f)",
    "r0_agrees_with_f(urn:case:stacks:f)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
    "proofGraph": {
      "nodes": [
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-1"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-1",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-10"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-10",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-11"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#enough_injectives"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-11",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-12"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#functor_between"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-12",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-13"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preserves_left_exactness"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-13",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#left_exact_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:LeftExactBySES:763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-13"
          ],
          "rule": "urn:stacks:clir:categories#LeftExactBySES",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:LeftExactIsAdditive:4450b8eb8e9d834d9659d2c1a4c4a9b8ab6cefd0744c6e9d16835294a6c3a258",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:LeftExactBySES:763a4180f5c5954ea9f59a77b201643905d37c183b5088a235f8e2a9b7e5fd41"
          ],
          "rule": "urn:stacks:clir:categories#LeftExactIsAdditive",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-14"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:s",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#short_exact_sequence"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-14",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-2"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-2",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-1",
            "urn:proof:assert:urn:mcp:case#fact-2"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByFiniteProducts",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-3"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_kernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-3",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-4"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_cokernels"
          },
          "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:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-5",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "definition": {
              "concept": "urn:stacks:clir:categories#abelian_category",
              "mode": "exact",
              "part": "sufficient"
            }
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#abelian_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a",
          "kind": "rule_application",
          "premises": [
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            "urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33",
            "urn:proof:assert:urn:mcp:case#fact-11",
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        },
        {
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    "schemaVersion": "law.core.evaluation/0.1"
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  "rulesApplied": [
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    "urn:stacks:clir:categories#LeftExactIsAdditive",
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    "urn:stacks:clir:derived-functors#DerivedFormDeltaFunctor",
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    "urn:stacks:clir:derived-functors#NegativeDerivedVanish",
    "urn:stacks:clir:derived-functors#R0AgreesIfLeftExact",
    "urn:stacks:clir:derived-functors#RFEverywhereDefined"
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}
SourcesExcerpts: 10

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
JSONRead only
{
  "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
JSONRead only
{
  "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
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      "contentHash": "sha256:73730814b6a7e9b8d49b7668a97550e999f7324072caaedeab6d267a7f2bbbe6",
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      "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}"
    }
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}

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
JSONRead only
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      "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
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    {
      "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
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  "contentHash": "sha256:75ae150b62d929f84cbd72f9c3c3bbc8b0e8a2fa9e10f51d42bf1c190b65d73e",
  "edition": "urn:stacks:clir:derived-functors#STACKS_DERIVED_MASTER",
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  "id": "urn:stacks:clir:derived-functors#ST_05TI",
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    {
      "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}"
    }
  ]
}

tag/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
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  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:categories#ST_00ZY",
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  "locator": "tag/00ZY",
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      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
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  "visibility": "public"
}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
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{
  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:categories#ST_0104",
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      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
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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",
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      "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}"
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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",
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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"
}

Packages in the snapshot

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
long_exact_sequence_of_derived: TRUE_ONLY — установлено Выведено правом: left_exact_functor(urn:case:stacks:f); additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); derived_functors_universal(urn:case:stacks:f); long_exact_sequence_of_derived(urn:case:stacks:f, urn:case:stacks:s); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f); r0_agrees_with_f(urn:case:stacks:f) Применены правила: AdditiveByFiniteProducts, LeftExactBySES, LeftExactIsAdditive, abelian_category/sufficient, DerivedFormDeltaFunctor, DerivedUniversalDeltaFunctor, LongExactSequence, NegativeDerivedVanish, R0AgreesIfLeftExact, RFEverywhereDefined Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 29 узлов — поле evaluation готово для law_explain

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  "codeHash": "sha256:9bcca6a33805c1c364ca1bc8e9d39d6a9c51ba44203c0406bafbf397b96be699",
  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317",
  "resultHash": "sha256:5a5a5e9ffbbb426e6eb0dca18926370eb2afc70ec3896e4e44411406c03fd3a8",
  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}
evaluation SHA-256
sha256:9394877720d8ad6bdf17cea824110ebe614591a1013a1bec9f4f55d8c847a9f2
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f",
    "urn:case:stacks:s"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "preserves_left_exactness"
    },
    {
      "args": [
        "urn:case:stacks:s",
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "short_exact_sequence"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "long_exact_sequence_of_derived",
  "proof": true
}

Короткая точная последовательность объектов порождает длинную точную последовательность производных.

Condition

Отрицательные степени нулевые

Context date 2026-09-06

Calculation result

Established

Input parameters

What we are finding

05TD (1): RiF=0R^iF = 0 for i<0i < 0

f

Input facts

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-a

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-a

  • all kernels exist in the category

    c: cat-a

  • all cokernels exist in the category

    c: cat-a

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-a

  • 00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: cat-b

  • 001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: cat-b

  • all kernels exist in the category

    c: cat-b

  • all cokernels exist in the category

    c: cat-b

  • the natural map Coim(f)→Im(f)Coim(f) → Im(f) is an isomorphism for all morphisms ff of the category

    c: cat-b

  • the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: cat-a

  • the functor goes from the first category to the second: F:A→BF : A → B

    f: fa: cat-ab: cat-b

  • F:Mor(x,y)→Mor(F(x),F(y))F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all objects x,yx, y

    f: f

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

Additional details

Include proof
Yes
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "homomorphism_on_hom_groups"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "negative_derived_functors_vanish",
  "proof": true
}
Why this resultApplied rules and conditions

Derivation path21 steps

  1. 1

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-a

    case fact
  2. 2

    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:cat-b

    case fact
  3. 3

    the category has enough injectives: every object AA has an injective morphism A→JA → J into an injective object JJ

    c: urn:case:stacks:cat-a

    case fact
  4. 4

    the functor goes from the first category to the second: F:A→BF : A → B

    f: urn:case:stacks:f; a: urn:case:stacks:cat-a; b: urn:case:stacks:cat-b

    case fact
  5. 5

    F:Mor(x,y)→Mor(F(x),F(y))F : Mor(x, y) → Mor(F(x), F(y)) is a homomorphism of abelian groups for all objects x,yx, y

    f: urn:case:stacks:f

    case fact
  6. 6

    00ZY: a functor of preadditive categories is additive if it is a homomorphism of abelian groups on morphism sets

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

    tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByHomGroups
    rule
  7. 7

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-a

    case fact
  8. 8

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-a

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  9. 9

    all kernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  10. 10

    all cokernels exist in the category

    c: urn:case:stacks:cat-a

    case fact
  11. 11

    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:cat-a

    case fact
  12. 12

    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:cat-a

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    rule
  13. 13

    00ZY: every Mor(x,y)Mor(x, y) is an abelian group and composition is bilinear

    c: urn:case:stacks:cat-b

    case fact
  14. 14

    001S: products x×yx × y exist for all pairs of objects, hence all finite products

    c: urn:case:stacks:cat-b

    case fact
  15. 15

    0104: a preadditive category with finite products is additive

    0104: the category is additive: c: urn:case:stacks:cat-b

    tag 0104, tag 00ZY

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts
    rule
  16. 16

    all kernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  17. 17

    all cokernels exist in the category

    c: urn:case:stacks:cat-b

    case fact
  18. 18

    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:cat-b

    tag 0109

    Identifier
    urn:stacks:clir:categories#abelian_category/sufficient
    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:f

    tag 05TI, tag 05SV, tag 05T4

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

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

    05TD (1): RiF=0R^iF = 0 for i<0i < 0: f: urn:case:stacks:f

    tag 05TD

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

    Query evaluation

    query

verified by the engine: 8 · case fact: 13 · Full graph: 24 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

  • 0104: a preadditive category with finite products is additive

    Identifier
    urn:stacks:clir:categories#AdditiveByFiniteProducts

  • 00ZY: a functor of preadditive categories is additive if it is a homomorphism of abelian groups on morphism sets

    Identifier
    urn:stacks:clir:categories#AdditiveByHomGroups

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

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

  • 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
Other rules in the evaluation1

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

Derived result for this query

  • 05TD (1): RiF=0R^iF = 0 for i<0i < 0

    f: f
Other derived facts7

  • the functor is additive

    f: f

  • 0104: the category is additive

    c: cat-a

  • 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: cat-a

  • 0104: the category is additive

    c: cat-b

  • 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: cat-b

  • 05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined

    f: f

  • 05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)

    f: f
the functor is additive
f
urn:case:stacks:f
0104: the category is additive
c
urn:case:stacks:cat-a
urn:case:stacks:cat-b
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:cat-a
urn:case:stacks:cat-b
05SV: the right derived functor RF:D⁺(A)→D⁺(B)RF : D⁺(A) → D⁺(B) is everywhere defined
f
f
05TE (1): the functors RiFR^iF , i≥0i ≥ 0 , carry a canonical structure of a cohomological δδ -functor (010Q)
f
f
05TD (1): RiF=0R^iF = 0 for i<0i < 0
f
f

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.

Proof graph

Proof graph · 6 layer
query_evaluationnegative_derived_functors_vanishrule_applicationNegativeDerivedVanishrule_applicationRFEverywhereDefinedrule_applicationAdditiveByHomGroupsrule_applicationabelian_category/sufficientrule_applicationabelian_category/sufficientassertionenough_injectivesassertionfunctor_betweenassertionhomomorphism_on_hom_groupsrule_applicationAdditiveByFiniteProductsassertionhas_all_kernelsassertionhas_all_cokernelsassertioncoimage_to_image_isomorphismrule_applicationAdditiveByFiniteProductsassertioncoimage_to_image_isomorphismassertionhas_all_kernelsassertionhas_all_cokernelsassertionpreadditive_categoryassertionhas_finite_productsassertionpreadditive_categoryassertionhas_finite_products

Proof nodes: 24 · assertion 13, rule_application 8, constraint_check 2, query_evaluation 1

assertion · urn:proof:assert:urn:mcp:case#fact-1
attributes
assertion
fact-1
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-1
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-10
attributes
assertion
fact-10
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-10
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-11
attributes
assertion
fact-11
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
enough_injectives
evidence
—
Identifier
fact-11
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-12
attributes
assertion
fact-12
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
  • Identifier
    cat-a
    Type
    entity_ref
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
functor_between
evidence
—
Identifier
fact-12
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-13
attributes
assertion
fact-13
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
homomorphism_on_hom_groups
evidence
—
Identifier
fact-13
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByHomGroups:7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_functor
evidence
—
Identifier
7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5
Type
rule_application
Premises
  • fact-13
Rule
AdditiveByHomGroups
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-2
attributes
assertion
fact-2
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-2
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
attributes
—
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
Type
rule_application
Premises
  • fact-1
  • fact-2
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-3
attributes
assertion
fact-3
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-3
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-4
attributes
assertion
fact-4
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-4
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-5
attributes
assertion
fact-5
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
coimage_to_image_isomorphism
evidence
—
Identifier
fact-5
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-a
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
Type
rule_application
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • fact-3
  • fact-4
  • fact-5
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-6
attributes
assertion
fact-6
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
preadditive_category
evidence
—
Identifier
fact-6
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-7
attributes
assertion
fact-7
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_finite_products
evidence
—
Identifier
fact-7
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
attributes
—
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
additive_category
evidence
—
Identifier
d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
Type
rule_application
Premises
  • fact-6
  • fact-7
Rule
AdditiveByFiniteProducts
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
assertion · urn:proof:assert:urn:mcp:case#fact-8
attributes
assertion
fact-8
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_kernels
evidence
—
Identifier
fact-8
Type
assertion
Premises
—
sourceAnchors
—
assertion · urn:proof:assert:urn:mcp:case#fact-9
attributes
assertion
fact-9
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
has_all_cokernels
evidence
—
Identifier
fact-9
Type
assertion
Premises
—
sourceAnchors
—
rule_application · urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
attributes
definition
concept
abelian_category
mode
exact
part
sufficient
conclusion
Arguments
  • Identifier
    cat-b
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
abelian_category
evidence
—
Identifier
df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
Type
rule_application
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • fact-10
  • fact-8
  • fact-9
Rule
abelian_category/sufficient
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:RFEverywhereDefined:f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
rf_everywhere_defined
evidence
—
Identifier
f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
Type
rule_application
Premises
  • 7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-11
  • fact-12
Rule
RFEverywhereDefined
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
v1
Identifier
cat-a
Type
entity_ref
v2
Identifier
cat-b
Type
entity_ref
rule_application · urn:proof:apply:DerivedFormDeltaFunctor:8f62e5cfbccfbc635c97b7ecd1e6ef2c33f53db9b99a642b5f0500ceabc5a9fd
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
derived_functors_form_delta_functor
evidence
—
Identifier
8f62e5cfbccfbc635c97b7ecd1e6ef2c33f53db9b99a642b5f0500ceabc5a9fd
Type
rule_application
Premises
  • f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
Rule
DerivedFormDeltaFunctor
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
rule_application · urn:proof:apply:NegativeDerivedVanish:9302e91a8e09a730e8bd3f7715567533ed4485085d1d6b6b087034c3cb4d8ff2
attributes
—
conclusion
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
negative_derived_functors_vanish
evidence
—
Identifier
9302e91a8e09a730e8bd3f7715567533ed4485085d1d6b6b087034c3cb4d8ff2
Type
rule_application
Premises
  • f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578
Rule
NegativeDerivedVanish
sourceAnchors
—
substitution
v0
Identifier
f
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
75c2ecfefcb4d5c2218bc87b7e9ccbfe0a659e05e4d62a89d32f01cb41f32484
Type
constraint_check
Premises
  • d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d
  • df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33
  • fact-10
  • fact-8
  • fact-9
sourceAnchors
—
substitution
v0
Identifier
cat-b
Type
entity_ref
constraint_check · urn:proof:constraint:abelian_category/necessary:ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
attributes
—
conclusion
constraint
abelian_category/necessary
requirementStatus
Established
Calculation status
Satisfied
triggerStatus
Satisfied
evidence
—
Identifier
ae494745d92071d97c9f5272871da104fa5cdf2902d5b41ac8f8584667e1432c
Type
constraint_check
Premises
  • d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662
  • d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a
  • fact-3
  • fact-4
  • fact-5
sourceAnchors
—
substitution
v0
Identifier
cat-a
Type
entity_ref
query_evaluation · urn:proof:query:mcp
attributes
—
conclusion
literal
Arguments
  • Identifier
    f
    Type
    entity_ref
Type
literal
Polarity
positive
Condition
negative_derived_functors_vanish
truthStatus
Established
evidence
—
Identifier
mcp
Type
query_evaluation
Premises
  • 9302e91a8e09a730e8bd3f7715567533ed4485085d1d6b6b087034c3cb4d8ff2
sourceAnchors
—
Calendar and proof identifiers
Proof reference
mcp
Original reasoning · JSON
JSONRead only
{
  "derived": [
    "additive_functor(urn:case:stacks:f)",
    "additive_category(urn:case:stacks:cat-a)",
    "abelian_category(urn:case:stacks:cat-a)",
    "additive_category(urn:case:stacks:cat-b)",
    "abelian_category(urn:case:stacks:cat-b)",
    "rf_everywhere_defined(urn:case:stacks:f)",
    "derived_functors_form_delta_functor(urn:case:stacks:f)",
    "negative_derived_functors_vanish(urn:case:stacks:f)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
    "proofGraph": {
      "nodes": [
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-1"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-1",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-10"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-10",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-11"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#enough_injectives"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-11",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-12"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              },
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#functor_between"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-12",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-13"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#homomorphism_on_hom_groups"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-13",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_functor"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByHomGroups:7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-13"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByHomGroups",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-2"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-2",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-1",
            "urn:proof:assert:urn:mcp:case#fact-2"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByFiniteProducts",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-3"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_kernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-3",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-4"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_cokernels"
          },
          "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:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#coimage_to_image_isomorphism"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-5",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "definition": {
              "concept": "urn:stacks:clir:categories#abelian_category",
              "mode": "exact",
              "part": "sufficient"
            }
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-a",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#abelian_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:AdditiveByFiniteProducts:d92ce9bcde5f250b0c0921a1ac5eae2b99e00c9dddc0db89c5753deea0ab2662",
            "urn:proof:assert:urn:mcp:case#fact-3",
            "urn:proof:assert:urn:mcp:case#fact-4",
            "urn:proof:assert:urn:mcp:case#fact-5"
          ],
          "rule": "urn:stacks:clir:categories#abelian_category/sufficient",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-6"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#preadditive_category"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-6",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-7"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:category-theory#has_finite_products"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-7",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#additive_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d",
          "kind": "rule_application",
          "premises": [
            "urn:proof:assert:urn:mcp:case#fact-6",
            "urn:proof:assert:urn:mcp:case#fact-7"
          ],
          "rule": "urn:stacks:clir:categories#AdditiveByFiniteProducts",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-8"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_kernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-8",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "assertion": "urn:mcp:case#fact-9"
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#has_all_cokernels"
          },
          "evidence": [],
          "id": "urn:proof:assert:urn:mcp:case#fact-9",
          "kind": "assertion",
          "premises": [],
          "sourceAnchors": []
        },
        {
          "attributes": {
            "definition": {
              "concept": "urn:stacks:clir:categories#abelian_category",
              "mode": "exact",
              "part": "sufficient"
            }
          },
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:cat-b",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:categories#abelian_category"
          },
          "evidence": [],
          "id": "urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:AdditiveByFiniteProducts:d62fcc6ab5f6182239b7f2500d945ab2594f951d60582596f4fc5d8fb83fba1d",
            "urn:proof:assert:urn:mcp:case#fact-10",
            "urn:proof:assert:urn:mcp:case#fact-8",
            "urn:proof:assert:urn:mcp:case#fact-9"
          ],
          "rule": "urn:stacks:clir:categories#abelian_category/sufficient",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:derived-functors#rf_everywhere_defined"
          },
          "evidence": [],
          "id": "urn:proof:apply:RFEverywhereDefined:f31eab7dcccab201e1dd3602b67595c12d19bf24acc845d662c662d0a5ef0578",
          "kind": "rule_application",
          "premises": [
            "urn:proof:apply:AdditiveByHomGroups:7b50108ebd390a92e5ff121e14523606441ac216c80c84f697f3cfd93e86f1a5",
            "urn:proof:apply:abelian_category/sufficient:d5030e23f7bbb2eb2d9287eb656bf463e4cc515b6281b0633e91e07fc93d248a",
            "urn:proof:apply:abelian_category/sufficient:df735a62aeff7766e9e4ef105380ce3f9dbfa080411bd695ec1e39f4b8ca6f33",
            "urn:proof:assert:urn:mcp:case#fact-11",
            "urn:proof:assert:urn:mcp:case#fact-12"
          ],
          "rule": "urn:stacks:clir:derived-functors#RFEverywhereDefined",
          "sourceAnchors": [],
          "substitution": {
            "v0": {
              "id": "urn:case:stacks:f",
              "kind": "entity_ref"
            },
            "v1": {
              "id": "urn:case:stacks:cat-a",
              "kind": "entity_ref"
            },
            "v2": {
              "id": "urn:case:stacks:cat-b",
              "kind": "entity_ref"
            }
          }
        },
        {
          "attributes": {},
          "conclusion": {
            "args": [
              {
                "id": "urn:case:stacks:f",
                "kind": "entity_ref"
              }
            ],
            "kind": "literal",
            "polarity": "positive",
            "predicate": "urn:stacks:clir:derived-functors#derived_functors_form_delta_functor"
          },
          "evidence": [],
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SourcesExcerpts: 8

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} .

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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",
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  "locator": "tag/05SV",
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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
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{
  "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
JSONRead only
{
  "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}"
    }
  ]
}

tag/00ZY

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 called preadditive if each morphism set Mor𝒜(x,y)\Mor_\mathcal{A}(x, y) is endowed with the structure of an abelian group such that the compositions

Mor(x,y)×Mor(y,z)⟶Mor(x,z)\Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z)

are bilinear. A functor F:𝒜→ℬF : \mathcal{A} \to \mathcal{B} of preadditive categories is called additive if and only if F:Mor(x,y)→Mor(F(x),F(y))F : \Mor(x, y) \to \Mor(F(x), F(y)) is a homomorphism of abelian groups for all x,y∈Ob(𝒜)x, y \in \Ob(\mathcal{A}) .

Original data · JSON
JSONRead only
{
  "contentHash": "sha256:600a2de3027b9725e89be51b838fa162e7bdc03f62971f32609663a9cc1b9bb7",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
  "fragmentKind": "defn",
  "id": "urn:stacks:clir:categories#ST_00ZY",
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  "locator": "tag/00ZY",
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    {
      "contentHash": "sha256:f2f20a29e596bff02bee9eee017d5b551092a1e17065b3a0638ec31466e411c6",
      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-preadditive}\nA category $\\mathcal{A}$ is called {\\it preadditive} if each\nmorphism set $\\Mor_\\mathcal{A}(x, y)$ is endowed\nwith the structure of an abelian group such that the\ncompositions\n$$\n\\Mor(x, y) \\times \\Mor(y, z)\n\\longrightarrow\n\\Mor(x, z)\n$$\nare bilinear. A functor $F : \\mathcal{A} \\to \\mathcal{B}$ of\npreadditive categories is called {\\it additive} if and only\nif $F : \\Mor(x, y) \\to \\Mor(F(x), F(y))$\nis a homomorphism of abelian groups for all\n$x, y \\in \\Ob(\\mathcal{A})$.\n\\end{definition}"
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  ],
  "visibility": "public"
}

tag/0104

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 called additive if it is preadditive and finite products exist, in other words it has a zero object and direct sums.

Original data · JSON
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  "contentHash": "sha256:2b8975d051697a9db95b3ff7f3fce2d90115db719d17e8fcc2f0682f0a58388a",
  "edition": "urn:stacks:clir:categories#STACKS_HOMOLOGY_MASTER",
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  "id": "urn:stacks:clir:categories#ST_0104",
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      "language": "en",
      "status": "official",
      "text": "\\begin{definition}\n\\label{definition-additive-category}\nA category $\\mathcal{A}$ is called {\\it additive}\nif it is preadditive and finite products exist, in other\nwords it has a zero object and direct sums.\n\\end{definition}"
    }
  ],
  "visibility": "public"
}

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",
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  "id": "urn:stacks:clir:categories#ST_0109",
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      "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}"
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}

Packages in the snapshot

  • Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — 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
Technical dataFull response, parameters and checksums
Calculation status
COMPUTED
Full engine response
negative_derived_functors_vanish: TRUE_ONLY — установлено Выведено правом: additive_functor(urn:case:stacks:f); additive_category(urn:case:stacks:cat-a); abelian_category(urn:case:stacks:cat-a); additive_category(urn:case:stacks:cat-b); abelian_category(urn:case:stacks:cat-b); rf_everywhere_defined(urn:case:stacks:f); derived_functors_form_delta_functor(urn:case:stacks:f); negative_derived_functors_vanish(urn:case:stacks:f) Применены правила: AdditiveByFiniteProducts, AdditiveByHomGroups, abelian_category/sufficient, DerivedFormDeltaFunctor, NegativeDerivedVanish, RFEverywhereDefined Право (вне юрисдикции государства): Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — доктрина (programHash sha256:b70b54495375…) 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 proof-граф: 24 узлов — поле evaluation готово для law_explain

Complete machine result · JSON

JSONRead only
{
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    "evaluationStatus": "COMPUTED",
    "kind": "TRUTH",
    "meaning": "установлено",
    "missingInputs": [],
    "truthStatus": "TRUE_ONLY"
  },
  "closedEditionRules": [],
  "derived": [
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    "abelian_category(urn:case:stacks:cat-a)",
    "additive_category(urn:case:stacks:cat-b)",
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    "rf_everywhere_defined(urn:case:stacks:f)",
    "derived_functors_form_delta_functor(urn:case:stacks:f)",
    "negative_derived_functors_vanish(urn:case:stacks:f)"
  ],
  "derivedOmitted": 0,
  "evaluation": {
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          "id": "urn:proof:query:mcp",
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  "vulnerableTo": [],
  "whyNot": []
}

Execution · JSON

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JSON · calculations, sources and exact data

JSONRead only
{
  "acts": [
    {
      "contributed": true,
      "fragmentCount": 16,
      "fragments": [
        "urn:stacks:clir:derived-functors#ST_05SV",
        "urn:stacks:clir:derived-functors#ST_05T4",
        "urn:stacks:clir:derived-functors#ST_05TD",
        "urn:stacks:clir:derived-functors#ST_05TE",
        "urn:stacks:clir:derived-functors#ST_05TI"
      ],
      "jurisdiction": "none",
      "namespace": "urn:stacks:clir:derived-functors",
      "package": "stacks-derived-functors",
      "title": "Производные функторы по The Stacks Project: определения и леммы как словарь с пошаговым раскрытием — вне юрисдикции государства — доктрина"
    },
    {
      "contributed": true,
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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"
    },
    {
      "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"
    }
  ],
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  "jurisdiction": "вне юрисдикции государства",
  "legalTime": "2026-09-06",
  "mode": "audit",
  "programHash": "sha256:b70b544953754dd80ba96365a1a8f6503235258ecd876a83a9d595d0e100f317",
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  "rustCodeHash": "sha256:d368cafc7162ed7a6563df5e5c943a57be26fa8aeb3179b530fe3bdd67fe9be4",
  "timezone": "Asia/Qyzylorda"
}
evaluation SHA-256
sha256:f720f268ab04ef90be173ad4cd34de1bd504ef30f160f18de99ad2dca6884c18
Original data · JSON
JSONRead only
{
  "args": [
    "urn:case:stacks:f"
  ],
  "facts": [
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "preadditive_category"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "has_finite_products"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_kernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "has_all_cokernels"
    },
    {
      "args": [
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-categories",
      "predicate": "coimage_to_image_isomorphism"
    },
    {
      "args": [
        "urn:case:stacks:cat-a"
      ],
      "package": "stacks-categories",
      "predicate": "enough_injectives"
    },
    {
      "args": [
        "urn:case:stacks:f",
        "urn:case:stacks:cat-a",
        "urn:case:stacks:cat-b"
      ],
      "package": "stacks-category-theory",
      "predicate": "functor_between"
    },
    {
      "args": [
        "urn:case:stacks:f"
      ],
      "package": "stacks-categories",
      "predicate": "homomorphism_on_hom_groups"
    }
  ],
  "kind": "truth",
  "legalTime": "2026-09-06",
  "package": "stacks-derived-functors",
  "predicate": "negative_derived_functors_vanish",
  "proof": true
}

Ниже нулевой степени производных функторов нет.

How to cite

The snapshot is immutable: the SHA-256 of the downloadable JSON pins it, so no access date is needed.

Citation
“При каких условиях правый производный функтор определён на всей категории, когда R⁰F совпадает с F и почему инъективный объект ацикличен?”. Arxo Lens, as of 2026-09-06. https://lens.arxo.io/a/a_yyr4HZvcrkQ5rn0nxn2fnKrs. Snapshot SHA-256: b649df74b1f0e98a92fadc11af0ad566d2f23b2acf34e1c772a79085f2cd3ba1.
BibTeX
@misc{arxo-lens-a_yyr4HZvcrkQ5,
  title = {При каких условиях правый производный функтор определён на всей категории, когда R⁰F совпадает с F и почему инъективный объект ацикличен?},
  howpublished = {Arxo Lens},
  url = {https://lens.arxo.io/a/a_yyr4HZvcrkQ5rn0nxn2fnKrs},
  note = {as of 2026-09-06; SHA-256 b649df74b1f0e98a92fadc11af0ad566d2f23b2acf34e1c772a79085f2cd3ba1}
}
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