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.

Ext from a two-term projective resolution

Example

Apply Hom(-,N) to a displayed two-term projective resolution and identify Ext^0 and Ext^1 as its kernel and cokernel, with higher groups zero.

Facts & Assumptions

Given: An exact sequence $0 \to P_{1} \overset{u}{\to} P_{0} \to M \to 0$ with $P_{0}, P_{1}$ projective, and an object $N$ .

Verification

technique · direct

1.1givenalgebra

Applying $Hom (-, N)$ gives the cochain complex $0 \to Hom (P_{0}, N) \overset{u^{*}}{\to} Hom (P_{1}, N) \to 0$ , where $u^{*} (f) = f \circ u$ . Its degree-zero kernel is $Hom (M, N)$ .

2.1step 1.1algebra∎

Consequently ${Ext}^{1} (M, N) = coker u^{*}$ , while ${Ext}^{q} (M, N) = 0$ for $q \geq 2$ because the displayed complex has no terms in those degrees. The projective-resolution computation is independent of this chosen two-term resolution.

Depends on

Used by

Ext of a cyclic abelian group by an abelian group Example

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

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 (standard reference, not scraped)