ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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.

A projective resolution of a cyclic abelian group

Example

For $n \geq 1$ , the cyclic abelian group $Z / n Z$ admits the projective resolution $0 \to Z \overset{n}{\to} Z \to Z / n Z \to 0,$ where the right-hand map is the quotient modulo $n$ .

Facts & Assumptions

Given: An integer $n \geq 1$ .

[L1]

Free modules are projective with the recorded choice boundary (Free modules are projective, with the exact choice boundary).

Verification

technique · direct

1.1

The composite of multiplication by $n$ with the quotient modulo $n$ is zero. The kernel of $Z \to Z / n Z$ is exactly $n Z$ , which is the image of the left map, and the left map is injective. So the sequence is exact.

givenalgebra

2.1

Both copies of $Z$ are free abelian groups of rank one and are therefore projective by [L1]. Hence the exact sequence of step 1.1 is a projective resolution of $Z / n Z$ .

L1step 1.1∎

Depends on

Free modules are projective, with the exact choice boundary

Used by

Two projective resolutions with nonisomorphic first syzygies Counterexample
Comparison maps between two resolutions of a cyclic group Example
The horseshoe resolution of an extension of cyclic groups Example

Dependency tree · two levels

5 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 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)