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

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Projective-resolution Ext has the stated bifunctor variance

Statement

Assume the Axiom of Dependent Choice. Let $A$ be an abelian category and let $P_{∙} (-)$ be supplied projective resolution data on every object of $A$ . For every $n \geq 0$ , ${Ext}_{P}^{n} (-, -) : A^{o p} \times A \to A b$ is contravariant in its first variable and covariant in its second variable.

Facts & Assumptions

Given: Morphisms $u : M^{'} \to M$ and $v : N \to N^{'}$ in $A$ .

Proof

technique · direct

1.1

Postcomposition with $v$ is a cochain map $Hom (P_{∙} (M), N) \to Hom (P_{∙} (M), N^{'})$ . A comparison lift $\tilde{u} : P_{∙} (M^{'}) \to P_{∙} (M)$ from A morphism has a comparison lift between the supplied projective resolutions gives precomposition ${\tilde{u}}^{*}$ in the opposite direction.

givenconstruct

2.1

These maps commute because pre- and postcomposition commute. Two lifts of $u$ are chain-homotopic by Projective comparison maps are unique up to chain homotopy. Precomposing with the homotopy gives a cochain homotopy between the two induced maps on $Hom (-, N)$ , so they induce the same map on $H^{n}$ . The comparison identity and composition laws hold up to such homotopy, while postcomposition is strictly functorial. Hence the maps on $H^{n}$ define the asserted bifunctor.

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

9 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)