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

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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

1.1L1L2L3construct

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.

2.1step 1.1∎

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

Depends on

Used by

Ext can be computed from any injective resolution of the second variable Corollary
Sheaf Ext of coherent modules Definition
The balanced Ext bifunctor Definition
A morphism has a comparison extension between the supplied injective resolutions Lemma
Long exact global sheaf Ext sequence in the first variable Lemma
Ext balance isomorphisms satisfy change-of-resolution cocycle laws Proposition
Grothendieck spectral sequence Theorem
Hyper-Ext spectral sequence Theorem
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)