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.

Projective dimension of a cyclic abelian group

Example

Use 0 -> Z --n--> Z -> Z/n -> 0 and the nonzero Ext^1 calculation to prove that Z/n has projective dimension one over Z.

Facts & Assumptions

Given: $M = Z / n Z$ for an integer $n \geq 2$ .

Verification

technique · direct

1.1givenconstruct

The exact sequence $0 \to Z \overset{n}{\to} Z \to M \to 0$ is a projective resolution of length one, so ${pd}_{Z} M \leq 1$ .

2.1step 1.1algebra∎

Applying ${Hom}_{Z} (-, Z)$ gives ${Ext}_{Z}^{1} (M, Z) ≅ Z / n Z \neq 0$ . The higher-Ext criterion therefore rules out projective dimension zero, so ${pd}_{Z} M = 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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, Chapters 3–4 (standard reference, not scraped)