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Two supplied injective resolution data define naturally isomorphic right derived functors

Statement

Assume the Axiom of Dependent Choice.

Let $I$ and $J$ be supplied injective resolution data on the same domain, and let $F : A \to B$ be an additive functor. For every $n \in Z$ , the additive functors $R_{I}^{n} F$ and $R_{J}^{n} F$ are naturally isomorphic.

Facts & Assumptions

Given: A morphism $u : A \to B$ and an integer $n$ .

[L1]

The two constructions define additive functors (Right derived functors relative to supplied data are additive functors).

[L2]

The chosen injective resolutions of a fixed object are homotopy equivalent under that object (Injective resolutions of the same object are homotopy equivalent under that object).

[L3]

Two injective comparison maps extending the same morphism are cochain-homotopic (Injective comparison maps are unique up to cochain homotopy).

[L4]

A cochain complex is read as a reindexed chain complex, and chain-homotopy invariance together with homology's compatibility with composition survives that reindexing (Cochain complex in an abelian category, Chain-homotopic maps induce the same map on homology, Homology respects identities and composition).

[L5]

Comparison extensions exist for morphisms on the supplied injective data (A morphism has a comparison extension between the supplied injective resolutions).

Proof

technique · direct

1.1

Fix an object $A$ . By [L2], there are comparison maps $c_{A}^{∙} : I^{∙} (A) \to J^{∙} (A)$ and $d_{A}^{∙} : J^{∙} (A) \to I^{∙} (A)$ whose composites are cochain- homotopic to the identities. Using [L4], these induce inverse isomorphisms $θ_{I, J} (A) : R_{I}^{n} F (A) \tilde{\to} R_{J}^{n} F (A) .$

L2L4givenconstruct

2.1

For a morphism $u : A \to B$ , choose comparison extensions ${\tilde{u}}^{I} : I^{∙} (A) \to I^{∙} (B)$ and ${\tilde{u}}^{J} : J^{∙} (A) \to J^{∙} (B)$ from [L5]. Both composites $c_{B}^{∙} {\tilde{u}}^{I}$ and ${\tilde{u}}^{J} c_{A}^{∙}$ extend $u$ , so [L3] makes them cochain-homotopic. By [L4], their induced cohomology maps agree, which is exactly the naturality square $θ_{I, J} (B) \circ R_{I}^{n} F (u) = R_{J}^{n} F (u) \circ θ_{I, J} (A) .$

L3L4L5step 1.1algebra

3.1

Steps 1.1 and 2.1 produce a natural isomorphism $R_{I}^{n} F \Rightarrow R_{J}^{n} F$ , and [L1] records that both sides are additive functors.

L1step 1.1step 2.1∎

Depends on

Used by

An acyclic object for a left exact functor Definition
Two resolution data and their change isomorphism Example
Change-of-injective-resolution isomorphisms satisfy identity and cocycle laws Proposition
Positive right derived functors vanish on injective objects Proposition
Derived functors are well defined relative to supplied resolution data Remark
The acyclic-resolution theorem for right derived functors Theorem

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Romyar Sharifi, Homological Algebra (standard reference, not scraped)