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

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Positive Tor vanishes when the resolved variable is projective

Statement

If the variable being resolved is projective, its resolution-defined ${Tor}_{i}^{R}$ vanishes for every $i > 0$ .

Proof

Given: a projective left module $M$ (or, symmetrically, a projective right module $N$ ).

1.1

Use the length-zero projective resolution $0 \to M \overset{1}{\to} M \to 0$ , concentrated in degree $0$ .

given

2.1

After tensoring with $N$ , the resulting complex is concentrated in degree $0$ with term $N \otimes_{R} M$ .

step 1.1algebra

3.1

Its homology in positive degrees is zero. Resolving a projective right module instead gives the symmetric conclusion.

step 2.1algebra∎

Depends on

Used by

Tor admits dimension shifting in either variable Proposition

Dependency tree · two levels

4 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)