CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

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Cohomology with a divisible abelian coefficient group is Hom of homology

Statement

If $C$ is a free abelian complex and $G$ is divisible, then $H^{n} {Hom}_{Z} (C, G) ≅ {Hom}_{Z} (H_{n} C, G)$ naturally.

Proof

Given: the cohomological UCT sequence and a divisible abelian group $G$ .

1.1given

A divisible abelian group is injective, so ${Ext}_{Z}^{1} (H_{n - 1} C, G) = 0$ .

2.1step 1.1∎

The UCT evaluation map consequently has zero kernel and is already surjective, and therefore is the stated natural isomorphism.

Depends on

Used by

Dependency tree · two levels

10 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)