ExampleConstruction: AI-generatedVerification: 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 of a cyclic abelian group by an abelian group

Example

Use the resolution 0 -> Z --n--> Z -> Z/n -> 0 to calculate Ext^1_Z(Z/n,A)=A/nA and show the higher terms vanish.

Facts & Assumptions

Given: An abelian group $A$ and an integer $n \geq 2$ .

Verification

technique · direct

1.1givenalgebra

The sequence $0 \to Z \overset{n}{\to} Z \to Z / n Z \to 0$ is a projective resolution: multiplication by $n$ is injective and its cokernel is $Z / n Z$ . Applying ${Hom}_{Z} (-, A)$ gives $0 \to A \overset{n}{\to} A \to 0$ .

2.1step 1.1algebra∎

Therefore ${Ext}_{Z}^{1} (Z / n Z, A) = coker (n : A \to A) = A / n A$ , and the same two-term complex has zero cohomology in every degree $q \geq 2$ .

Depends on

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

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