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.

Results

Condition

RF определён на всей категории

Established

Query parameters · 1

f: urn:case:stacks:f

Input facts · 13

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-b
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-b
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the functor goes from the first category to the second: $F : A → B$
f: fa: cat-ab: cat-b
$F : Mor(x, y) → Mor(F(x), F(y))$ is a homomorphism of abelian groups for all objects $x, y$
f: f

Why this result? →Sources: 8

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

Condition

Инъективная резольвента существует

Established

Query parameters · 1

a: urn:case:stacks:x

Input facts · 7

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the object belongs to the category: $x ∈ Ob(C)$
x: xc: cat-a

Why this result? →Sources: 4

Достаточность инъективных объектов даёт резольвенту для каждого объекта категории.

Condition

R¹ по инъективной резольвенте

Established

Query parameters · 4

f: urn:case:stacks:f
n: 1
a: urn:case:stacks:x
h: urn:case:stacks:h1

Input facts · 21

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-b
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-b
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the functor goes from the first category to the second: $F : A → B$
f: fa: cat-ab: cat-b
$F : Mor(x, y) → Mor(F(x), F(y))$ is a homomorphism of abelian groups for all objects $x, y$
f: f
the object belongs to the category: $x ∈ Ob(C)$
x: xc: cat-a
$K^•$ is a complex in the category
k: ic: cat-a
013I (1): $I^n = 0$ for $n < 0$
k: i
013I (2): each $I^n$ is an injective object of the category
k: i
013I (3): the map $A → I^0$ is an isomorphism onto $Ker(d^0)$
a: xk: i
013I (4): $H^i(I^•) = 0$ for $i > 0$
k: i
$F(K^•)$ is the given complex: $F$ applied termwise to $K^•$
f: fk: ifk: f-i
$H^n(K^•)$ is the given object
k: f-in: 1h: h1

Why this result? →Sources: 13

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

Condition

Инъективный объект ацикличен

Established

Query parameters · 2

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

Input facts · 15

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-b
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-b
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the functor goes from the first category to the second: $F : A → B$
f: fa: cat-ab: cat-b
for every short exact sequence $0 → A → B → C → 0$ the sequence $0 → F(A) → F(B) → F(C)$ is exact
f: f
the object belongs to the category: $x ∈ Ob(C)$
x: jc: cat-a
for every injection $A ↪ B$ and every morphism $A → J$ there exists a morphism $B → J$ making the diagram commute
j: j

Why this result? →Sources: 14

На инъективном объекте производные положительной степени обращаются в нуль.

Condition

R⁰F совпадает с F при левой точности

Established

Query parameters · 1

f: urn:case:stacks:f

Input facts · 13

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-b
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-b
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the functor goes from the first category to the second: $F : A → B$
f: fa: cat-ab: cat-b
for every short exact sequence $0 → A → B → C → 0$ the sequence $0 → F(A) → F(B) → F(C)$ is exact
f: f

Why this result? →Sources: 10

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

Condition

Длинная точная последовательность

Established

Query parameters · 2

f: urn:case:stacks:f
s: urn:case:stacks:s

Input facts · 14

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-b
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-b
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the functor goes from the first category to the second: $F : A → B$
f: fa: cat-ab: cat-b
for every short exact sequence $0 → A → B → C → 0$ the sequence $0 → F(A) → F(B) → F(C)$ is exact
f: f
the sequence $0 → A → B → C → 0$ is a short exact sequence in the category
s: sc: cat-a

Why this result? →Sources: 10

Короткая точная последовательность объектов порождает длинную точную последовательность производных.

Condition

Отрицательные степени нулевые

Established

Query parameters · 1

f: urn:case:stacks:f

Input facts · 13

00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-a
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-a
00ZY: every $Mor(x, y)$ is an abelian group and composition is bilinear
c: cat-b
001S: products $x × y$ 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 $Coim(f) → Im(f)$ is an isomorphism for all morphisms $f$ of the category
c: cat-b
the category has enough injectives: every object $A$ has an injective morphism $A → J$ into an injective object $J$
c: cat-a
the functor goes from the first category to the second: $F : A → B$
f: fa: cat-ab: cat-b
$F : Mor(x, y) → Mor(F(x), F(y))$ is a homomorphism of abelian groups for all objects $x, y$
f: f

Why this result? →Sources: 8

Ниже нулевой степени производных функторов нет.

How to cite

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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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