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.

Depends on

Used by

The balanced Tor bifunctor Definition
The Tor balance isomorphism is natural and coherent under a change of resolutions Proposition
The long exact Tor sequence in the left-module variable Theorem
The long exact Tor sequence in the right-module variable Theorem
Tor is symmetric over a commutative ring Theorem
Tor one of a cyclic abelian group detects n-torsion Theorem

Dependency tree · two levels

23 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

Weibel, An Introduction to Homological Algebra, Comparison Theorem 2.2.6 and Lemma 2.4.4/Theorem 2.4.5, pp.35–36 and 44–45 (standard reference, not scraped)
Charles A. Weibel, An Introduction to Homological Algebra (standard reference, not scraped)