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

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Kunneth over a field

Statement

Assume the Axiom of Choice. For complexes of vector spaces over a field $k$ with finite diagonals, cross product is a natural isomorphism $⨁_{p + q = n} H_{p} C \otimes_{k} H_{q} D ≅ H_{n} (C \otimes_{k} D)$ .

Proof

Given: The Axiom of Choice (The Axiom of Choice) and the Kunneth exact sequence over $k$ .

1.1given

Under the stated Choice hypothesis, every $k$ -module is flat by Modules over a field are projective, flat, and injective, hence ${Tor}_{1}^{k} (H_{p} C, H_{q} D) = 0$ for all $p, q$ .

2.1givenstep 1.1∎

Under the same Choice hypothesis, The Kunneth theorem for free complexes over a PID supplies the natural exact sequence. Its Tor surjection has zero target by step 1.1, while cross product remains injective, giving the stated natural isomorphism.

Depends on

Dependency tree · 2 levels

Sources

Weibel, An Introduction to Homological Algebra