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

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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.
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The augmented rows of the tensor double complex are exact

Statement

If $Q_{∙} ↠ N$ and $P_{∙} ↠ M$ are projective resolutions, every augmented row $Q_{p} \otimes_{R} P_{∙} \to Q_{p} \otimes_{R} M$ is exact.

Proof

Given: a fixed projective right module $Q_{p}$ and the augmented resolution $P_{∙} \to M$ .

1.1

The module $Q_{p}$ is flat as a right module, so $Q_{p} \otimes_{R} -$ preserves the exact augmented complex $P_{∙} \to M$ .

given

2.1

Its degree- $q$ terms are exactly $Q_{p} \otimes_{R} P_{q}$ , and its augmentation is $Q_{p} \otimes_{R} P_{0} \to Q_{p} \otimes_{R} M$ .

step 1.1algebra

3.1

Thus the row indexed by $p$ is exact, including the augmentation; this is the claimed augmented-row condition.

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)