Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)
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‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Freeness of chain groups cannot simply be dropped from the classical Kunneth statement

Statement

The classical free-complex Kunneth argument cannot simply omit freeness of chain groups: it uses freeness to make cycle and boundary modules flat and to control the Tor edge.

Refutation

Given: the complexes C=D=Z/2 concentrated in degree zero, viewed as complexes of nonfree abelian groups.

1.1

Their ordinary tensor complex is concentrated in degree zero, so H1(C⊗ZD)=0. On the other hand, H0C=H0D=Z/2 and Tor⁡1Z(H0C,H0D)=Z/2.

given
2.1

If the classical free-complex Kunneth short exact sequence were asserted unchanged after simply deleting freeness, then in total degree one it would surject from the zero group H1(C⊗D) onto the nonzero Tor group from step 1.1. That is impossible. Hence freeness cannot simply be dropped without replacing ordinary tensor by a derived construction or adding suitable flatness hypotheses.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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