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.

An Ext dimension shift

Example

Use the displayed cyclic-group projective presentation to identify Ext^{n+1}(M,N) with Ext^n(Omega M,N), distinguishing the low-degree exact segment.

Facts & Assumptions

Given: $M = Z / n Z$ with $n \geq 2$ , its presentation $0 \to Z \overset{n}{\to} Z \to M \to 0$ , and an abelian group $N$ .

Verification

technique · direct

1.1givenalgebra

The first syzygy of $M$ for this presentation is $Ω M ≅ Z$ , which is free and hence projective. The low-degree portion of the long exact sequence is $0 \to Hom (M, N) \to N \overset{n}{\to} N \to {Ext}^{1} (M, N) \to 0$ .

2.1step 1.1algebra∎

For every $q \geq 1$ , dimension shifting gives ${Ext}^{q + 1} (M, N) ≅ {Ext}^{q} (Z, N) = 0$ . This does not replace the displayed low-degree segment: its cokernel is the generally nonzero group $N / n N = {Ext}^{1} (M, N)$ .

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, Chapter 2 (standard reference, not scraped)