PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-09 (gpt-6-astra)

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Ext balance isomorphisms satisfy change-of-resolution cocycle laws

Statement

Assume Dependent Choice (DC) in an abelian category with enough projectives and injectives, and supply the projective and injective resolutions used below. For fixed objects $M, N$ and $n \geq 0$ , write $E_{P} = {Ext}_{P}^{n} (M, N)$ and $E_{I} = {Ext}_{I}^{n} (M, N)$ . Let $c_{P, P^{'}} : E_{P} \to E_{P^{'}}$ and $c_{I, I^{'}} : E_{I} \to E_{I^{'}}$ be the maps induced by resolution comparisons over or under the identity objects. Then $c_{P, P} = 1, c_{P^{'}, P^{''}} c_{P, P^{'}} = c_{P, P^{''}}, c_{I, I} = 1, c_{I^{'}, I^{''}} c_{I, I^{'}} = c_{I, I^{''}} .$ These maps are isomorphisms. For the balance maps $β_{P, I} : E_{P} \to E_{I}$ one has the correctly typed square $c_{I, I^{'}} β_{P, I} = β_{P^{'}, I^{'}} c_{P, P^{'}} .$ Thus successive changes through any intermediate supplied resolutions give the same identification as direct comparison, naturally in both objects. The identity laws concern unchanged presentations; $β_{P, I}$ between different presentations is not asserted to be a literal identity.

Facts & Assumptions

Given: the stated resolution data and The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain. The balance maps are supplied by Projective and injective constructions of Ext agree for supplied resolutions.

Proof

technique · direct

1.1givenconstruct

By Projective comparison maps exist choose $f : P^{'} \to P$ lifting $1_{M}$ , and let $c_{P, P^{'}} = H^{n} (f^{*})$ on the Hom cochain complexes. Two such $f, g$ differ by $d h + h d$ by Projective comparison maps are unique up to chain homotopy. Precomposition by $h$ , in degree $n$ sending $ϕ$ to $ϕ h_{n - 1}$ , is a cochain homotopy between $f^{*}$ and $g^{*}$ , so their induced maps on cohomology coincide. Identity comparison gives $c_{P, P} = 1$ . A composite $P^{''} \to P^{'} \to P$ lifts $1_{M}$ , so comparison uniqueness proves $c_{P^{'}, P^{''}} c_{P, P^{'}} = c_{P, P^{''}}$ . Opposite-direction comparisons consequently induce inverse maps.

2.1givenconstruct

By Injective comparison maps exist choose $g : I \to I^{'}$ extending $1_{N}$ and let $c_{I, I^{'}} = H^{n} (g_{*})$ . The uniqueness theorem Injective comparison maps are unique up to cochain homotopy supplies a homotopy between any two such extensions; postcomposition with it is a Hom cochain homotopy. Thus this map is independent of the extension. Identity extensions and composites give the two injective comparison laws, and reverse extensions give inverses, exactly as in step 1.1.

3.1step 1.1step 2.1algebra∎

The chosen comparison maps induce the augmentation-compatible Hom double-complex squares in The Ext balance isomorphism is independent of resolution comparison data. Both edge-to-total maps commute with them. Taking cohomology and inverting the appropriate edge isomorphism proves $c_{I, I^{'}} β_{P, I} = β_{P^{'}, I^{'}} c_{P, P^{'}}$ . The comparison laws of steps 1.1–2.1 and this square reduce successive changes, including mixed projective/injective changes, to direct comparison. Naturality in module morphisms follows from The Ext balance isomorphism is natural in both variables and comparison uniqueness: the two composite lifts in each change-of-resolution naturality square lift the same object morphism and hence induce equal cohomology maps.

Remarks

DC supplies arbitrary comparisons and their homotopies. The equations are choice-free when these data are explicitly supplied. This proof does not use a uniqueness theorem for arbitrary morphisms of delta functors merely because they agree in degree zero.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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)