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.

Degree-zero Tor is the tensor product in either construction

Statement

For a right $R$ -module $N$ and a left $R$ -module $M$ , both resolution constructions give ${Tor}_{0}^{R} (N, M) ≅ N \otimes_{R} M$ .

Proof

Given: a projective resolution $P_{∙} ↠ M$ and a projective resolution $Q_{∙} ↠ N$ .

1.1

The augmented complex $P_{1} \to P_{0} \to M \to 0$ remains right exact after $N \otimes_{R} -$ , so $H_{0} (N \otimes_{R} P_{∙}) = coker (N \otimes P_{1} \to N \otimes P_{0})$ .

given

2.1

That cokernel is $N \otimes_{R} M$ by the displayed right-exact sequence, so the left-resolved construction has the asserted degree-zero value.

step 1.1algebra

3.1

The same calculation for $Q_{∙} \otimes_{R} M$ gives $coker (Q_{1} \otimes M \to Q_{0} \otimes M) = N \otimes_{R} M$ , proving both identifications.

step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)