PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)verified 2026-09-09 (gpt-6-astra)

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Each resolution-defined Tor construction is covariant in both variables

Statement

Assume Dependent Choice (DC). For modules equipped with projective resolutions, the homology groups obtained by resolving either variable are covariantly functorial in the right module $N$ and the left module $M$ . Maps do not depend on the chosen comparison lifts. This assertion uses supplied resolutions; it does not assert that DC supplies projective resolutions of every module.

Proof

Given: module maps $u : N \to N^{'}$ and $v : M \to M^{'}$ , supplied projective resolutions, and DC.

Facts & Assumptions. The two homology constructions are Tor from a projective resolution of the left module and Tor from a projective resolution of the right module. Under The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain, Projective comparison maps exist supplies comparison maps and Projective comparison maps are unique up to chain homotopy supplies homotopies between lifts of the same module map.

1.1givenconstruct

DC supplies a comparison map $f : P_{∙} \to P_{∙}^{'}$ lifting $v$ . The tensor map $u \otimes f$ gives the map on the left-resolution complexes, hence on homology. DC is used only in the countable compatible selections of comparison and homotopy components supplied by the cited theorems.

2.1step 1.1algebra

If $f - g = d h + h d$ , tensoring gives $u \otimes (f - g) = d (u \otimes h) + (u \otimes h) d$ . On a cycle the difference is therefore a boundary. Thus the induced homology map is independent of the comparison lift.

3.1step 1.1step 2.1algebra∎

Identity chain maps lift identities, and a composite of lifts lifts the composite module map. Any other lift is homotopic to it, so step 2.1 gives the identity and composition laws. Tensor maps in different variables commute on pure tensors and hence on the whole tensor product. For the right-resolution construction lift $u$ to $g : Q \to Q^{'}$ and use $g \otimes v$ ; its homotopies and the same computations give both functor laws.

Remarks

For explicitly supplied chain maps and homotopies, the tensor and homology calculations above are choice-free. Without DC, a functorial interpretation may instead be made conditional on supplied lifts and homotopies witnessing their independence, identities and compositions; these data are not silently assumed to exist.

Each resolution-defined Tor construction is covariant in both variables

Statement

Proof

Remarks

Depends on

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