ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 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.

Universal-coefficient cohomology of a two-term free complex

Statement

For a nonzero integer $m$ , let $C = (0 \to Z \overset{m}{\to} Z \to 0)$ . With coefficients $Z$ , one has $H^{0} Hom (C, Z) = 0$ and $H^{1} Hom (C, Z) ≅ Z / m$ .

Verification

Given: the cochain complex $0 \leftarrow Z \overset{m}{\leftarrow} Z \leftarrow 0$ .

1.1given

Multiplication by nonzero $m$ on $Z$ has kernel $0$ and cokernel $Z / m$ .

2.1step 1.1∎

Thus the cochain calculation has the displayed cohomology; it agrees with ${Ext}_{Z}^{1} (Z / m, Z) ≅ Z / m$ .

Depends on

The universal coefficient theorem for cohomology over a PID

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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)