TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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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.

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Injective comparison maps exist

Statement

Assume the Axiom of Dependent Choice.

Let $u : A \to B$ be a morphism, and let $I^{∙}$ and $J^{∙}$ be injective resolutions of $A$ and $B$ . Then there exists a coaugmentation-preserving cochain map $I^{∙} \to J^{∙}$ extending $u$ .

Facts & Assumptions

Given: A morphism $u : A \to B$ and injective resolutions $I^{∙}$ of $A$ and $J^{∙}$ of $B$ .

[L1]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

[L2]

An injective resolution is the cochain datum to be dualized (Injective resolutions in an abelian category).

[L3]

Projective comparison maps exist (Projective comparison maps exist).

Proof

technique · direct

1.1

By [L1], pass to the opposite abelian category. Reversing arrows turns the given injective resolutions from [L2] into projective resolutions there, and $u : A \to B$ becomes a morphism in the opposite direction. Apply [L3] in the opposite category to obtain the required comparison map.

L1L2L3construct

2.1

Translating that chain map back to the original category reverses arrows again and yields a cochain map $I^{∙} \to J^{∙}$ extending $u$ .

step 1.1∎

Depends on

Used by

Injective resolutions of the same object are homotopy equivalent under that object Theorem

Dependency tree · two levels

14 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

Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)