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

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The augmented fixed-q rows of the tensor double complex are exact

Statement

For the same two projective resolutions, every augmented fixed- $q$ row $Q_{∙} \otimes_{R} P_{q} \to N \otimes_{R} P_{q}$ is exact.

Proof

Given: a fixed projective left module $P_{q}$ and the augmented resolution $Q_{∙} \to N$ .

1.1

The module $P_{q}$ is flat as a left module, so $- \otimes_{R} P_{q}$ preserves exactness of $Q_{∙} \to N$ .

given

2.1

The resulting degree- $p$ terms are $Q_{p} \otimes_{R} P_{q}$ , with augmentation $Q_{0} \otimes_{R} P_{q} \to N \otimes_{R} P_{q}$ .

step 1.1algebra

3.1

Hence every augmented fixed- $q$ row is exact, on the correct right-module/left-module tensor convention.

step 2.1algebra∎

Depends on

Used by

The left and right projective constructions of Tor are naturally isomorphic Theorem

Dependency tree · two levels

7 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)