TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06‡ rests on unproved material (inherited)

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.

‡ 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.

The Kunneth theorem for free complexes over a PID

Statement

Let $R$ be a PID and $C, D$ be complexes of free $R$ -modules for which each total degree has a finite direct-sum diagonal. There is a natural exact sequence $0 \to ⨁_{p + q = n} H_{p} C \otimes_{R} H_{q} D \to H_{n} (C \otimes_{R} D) \to ⨁_{p + q = n - 1} {Tor}_{1}^{R} (H_{p} C, H_{q} D) \to 0$ .

Proof

Given: free PID-complexes $C, D$ with finite direct-sum diagonal in every total degree.

1.1

Every $C_{p}$ is free, and every boundary module $d (C_{p}) = B_{p - 1} C$ is free by Boundaries and cycles in a free complex over a PID are free; hence all of these modules are flat.

given

2.1

Weibel's cited Kunneth formula for complexes applies to the right complex $C$ and left complex $D$ under exactly the flatness conditions verified in step 1.1. It gives the displayed natural short exact sequence; its left map is the cross product of The Kunneth cross-product map is well defined and natural, and its right map is the quotient of The Kunneth Tor map. The finite-diagonal hypothesis makes each displayed direct sum finite; an empty diagonal gives the zero module.

step 1.1algebra∎

Depends on

Used by

Kunneth over a field Corollary
Kunneth when one homology family is flat Corollary
Kunneth for two cyclic two-term complexes Example
Freeness of chain groups cannot simply be dropped from the classical Kunneth statement False statement
Kunneth over a PID is not always a tensor-product isomorphism False statement
Euler characteristic is multiplicative under the finite Kunneth hypotheses Proposition
The Kunneth sequence splits nonnaturally Theorem

Dependency tree · two levels

13 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, Theorem 3.6.3, printed p. 88 (standard reference, not scraped)