PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Each resolution-defined Tor construction is covariant in both variables

Statement

The homology groups obtained by resolving either variable define covariant functors of the right module $N$ and the left module $M$ .

Proof

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

1.1

A comparison map lifting $v$ is a chain map $P_{∙} \to P_{∙}^{'}$ ; tensoring with $N$ gives a chain map $N \otimes P_{∙} \to N \otimes P_{∙}^{'}$ , while $u \otimes 1$ handles the first variable.

given

2.1

Chain-homotopic comparison maps induce the same map on homology, so the map of $Tor$ does not depend on the chosen lift of $v$ .

step 1.1algebra

3.1

Composition of comparison maps is chain-homotopic to a comparison map for the composite, hence identity and composition laws hold on homology; the right-resolved construction is identical with the variables interchanged.

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

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 (standard reference, not scraped)