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

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Kunneth when one homology family is flat

Statement

Assume the Axiom of Choice. Under the free-chain and finite-diagonal Kunneth hypotheses, if every $H_{p} C$ is flat then cross product is a natural isomorphism.

Proof

Given: The Axiom of Choice (The Axiom of Choice), free PID-complexes $C, D$ with finite diagonals, and flatness of every $H_{p} C$ .

1.1given

Under Choice, the free cycle-boundary presentation $0 \to B_{q} D \to Z_{q} D \to H_{q} D \to 0$ is a supplied projective resolution by Boundaries and cycles in a free complex over a PID are free. Choice implies dependent choice: for a serial relation, choose one successor for each state using Choice and iterate that successor by recursion on $N$ . Thus the cited flatness/Tor criterion applies to this resolution and gives ${Tor}_{1}^{R} (H_{p} C, H_{q} D) = 0$ for every pair $(p, q)$ .

2.1step 1.1given∎

The Kunneth sequence in The Kunneth theorem for free complexes over a PID has a finite direct sum of these correction terms in each degree. It is zero by step 1.1, so exactness makes the natural cross-product injection surjective as well.

Depends on

Used by

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Dependency tree · two levels

17 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)