DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedaudited 2026-09-06

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.

The evaluation map from cohomology to Hom of homology

Definition

Let $C$ be a chain complex of $R$ -modules and let $G$ be an $R$ -module. For every $n \in Z$ , define the evaluation map ${ev}_{n} : H^{n} {Hom}_{R} (C, G) ⟶ {Hom}_{R} (H_{n} C, G), {ev}_{n} ([f]) ([z]) = f (z) .$ A cocycle $f : C_{n} \to G$ vanishes on $B_{n} C$ , so its value depends only on $[z]$ ; changing $f$ by a coboundary does not change its value on cycles. Thus the displayed formula is well defined.

Depends on

Used by

The universal coefficient theorem for cohomology over a PID Theorem

Dependency tree · two levels

7 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)