PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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Euler characteristic is multiplicative under the finite Kunneth hypotheses

Statement

Assume the Axiom of Choice. Let $R$ be a PID and let $C, D$ be complexes of free $R$ -modules with finite total-degree diagonals. If their homology modules have finite rank and only finitely many are nonzero, then $χ (C \otimes_{R} D) = χ (C) χ (D)$ .

Proof

Given: the stated PID, freeness, finite-diagonal, finite-homology, and Axiom-of-Choice hypotheses (The Axiom of Choice).

1.1givenalgebra

The exact sequence of The Kunneth theorem for free complexes over a PID applies under the stated choice and finite-diagonal hypotheses. Tensor it with the fraction field $K$ of $R$ ; Localisation of modules is exact preserves exactness and homology. A free resolution used to compute ${Tor}_{1}^{R} (H_{p} C, H_{q} D)$ localizes to a free resolution over the field $K$ , so its degree-one homology vanishes. Tensor associativity identifies $(H_{p} C \otimes_{R} H_{q} D) \otimes_{R} K$ with $(H_{p} C \otimes_{R} K) \otimes_{K} (H_{q} D \otimes_{R} K)$ . Thus $rank H_{n} (C \otimes_{R} D) = \sum_{p + q = n} rank H_{p} C rank H_{q} D$ .

2.1step 1.1algebra∎

Taking the finite alternating sum and regrouping it gives $\sum_{p, q} (- 1)^{p + q} rank H_{p} C rank H_{q} D = χ (C) χ (D)$ .

Depends on

Used by

Euler characteristic of a tensor-product complex Example

Dependency tree · two levels

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Sources

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