PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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Injective-resolution Ext has the stated bifunctor variance

Statement

Assume the Axiom of Dependent Choice. Let $I$ be supplied injective resolution data on a class $D$ in an abelian category. For every $q$ , injective-resolution Ext is contravariant in $M$ and covariant in $N$ : $a : M^{'} \to M$ and $b : N \to N^{'}$ induce ${Ext}_{I}^{q} (M, N) \to {Ext}_{I}^{q} (M^{'}, N)$ and ${Ext}_{I}^{q} (M, N) \to {Ext}_{I}^{q} (M, N^{'})$ .

Facts & Assumptions

Given: Objects $N, N^{'} \in D$ , objects $M, M^{'}$ , and maps $a, b$ as stated.

Proof

technique · direct

1.1

Precomposition by $a$ is a cochain map $Hom (M, I^{∙}) \to Hom (M^{'}, I^{∙})$ . A comparison extension of $b$ is a cochain map $I^{∙} \to I^{' ∙}$ , hence postcomposition gives the second cochain map.

givenconstruct

2.1

Homotopic comparison extensions induce the same cohomology map, so the second map is choice-independent. Identity and composition of the comparison maps give the functor laws, with $a$ reversing arrows.

step 1.1algebra∎

Depends on

Used by

FALSE: Ext is covariant in both variables False statement
The Ext balance isomorphism is natural in both variables Proposition

Dependency tree · two levels

12 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, Chapter 2 (standard reference, not scraped)