TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1L2L4givenconstruct

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 $θ_{I, J} (A) : R_{I}^{n} F (A) \tilde{\to} R_{J}^{n} F (A) .$

2.1L3L4L5step 1.1algebra

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 $θ_{I, J} (B) \circ R_{I}^{n} F (u) = R_{J}^{n} F (u) \circ θ_{I, J} (A) .$

3.1L1step 1.1step 2.1∎

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.

Depends on

Dependency tree · 2 levels

Sources

Romyar Sharifi, Homological Algebra