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False statementConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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The homological and cohomological UCT correction terms are not reversed

Statement

Assume the Axiom of Choice. The asserted reversal of the UCT correction terms is false: for free chain complexes over a PID, homology has Tor⁡1 and cohomology has Ext⁡1.

Refutation

Given: the Axiom of Choice and the two AC-qualified UCT exact sequences.

1.1given

The homological UCT The universal coefficient theorem for homology over a PID for a free PID chain complex places Tor⁡1(Hn−1C,G) on the quotient side of its exact sequence.

2.1step 1.1∎

The cohomological UCT The universal coefficient theorem for cohomology over a PID places Ext⁡1(Hn−1C,G) on the kernel side of its exact sequence. Thus the asserted reversal is false under the same Choice premise.

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