DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

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.

The tensor product of a right and a left chain complex is totalized on finite diagonals with the Koszul differential

Definition

Let $R$ be a ring, $P$ a chain complex of right $R$ -modules and $Q$ a chain complex of left $R$ -modules. Define $(P \otimes_{R} Q)_{n} = ⨁_{p + q = n} P_{p} \otimes_{R} Q_{q}$ (a finite-diagonal direct sum) and $d (p \otimes q) = d_{P} p \otimes q + (- 1)^{p} p \otimes d_{Q} q$ .

Depends on

Used by

A chain complex with coefficients obtained by tensoring Definition
The first-quadrant tensor double complex of two projective resolutions Definition
The homology cross product for tensor complexes Definition
Tor from a projective resolution of the left module Definition
Tor from a projective resolution of the right module Definition
The tensor-total differential is balanced, well defined, and squares to zero Lemma
The Kunneth theorem for free complexes over a PID Theorem

Dependency tree · two levels

8 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

Weibel, An Introduction to Homological Algebra (standard reference, not scraped)