False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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The two resolution constructions of Tor are not equal by definition

Statement

False claim: resolving the left and resolving the right variable produce literally equal Tor complexes by definition.

Refutation

Given: $R = Z$ , $N = Z / 2$ , $M = Z / 3$ , and their standard two-term free resolutions.

1.1given

Resolving $M$ gives the left-resolved complex $Z / 2 \overset{3}{\to} Z / 2$ , whose differential is the identity. Resolving $N$ gives the right-resolved complex $Z / 3 \overset{2}{\to} Z / 3$ .

2.1step 1.1algebra

These complexes are not literally equal: even their degree-zero groups are $Z / 2$ and $Z / 3$ . Nevertheless both have zero homology, as required because $Z / 2 \otimes Z / 3 = 0$ and the positive Tor groups also vanish.

3.1step 2.1algebra∎

The tensor double-complex argument supplies a natural isomorphism only after passing to homology; this refutes equality by definition.

Depends on

The left and right projective constructions of Tor are naturally isomorphic

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)