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.

A noncommutative Yoneda product

Example

Use a finite-dimensional algebra with composable nonsymmetric extension classes to calculate two products in opposite orders and show they differ or one is undefined by endpoints.

Facts & Assumptions

Given: A field $k$ , $R = k ⟨ x, y ⟩ / (x, y)^{2}$ , and the simple left $R$ -module $S = R / (x, y)$ .

Verification

technique · direct

1.1givenconstruct

Let $V = (x, y)$ , so $V^{2} = 0$ . The free resolution $P_{n} = R \otimes_{k} V^{\otimes n}$ with differential given by multiplication by the first $V$ -factor has $\ker d_{n} = V \otimes_{k} V^{\otimes n} = im d_{n + 1}$ . Applying ${Hom}_{R} (-, S)$ gives zero differentials, so ${Ext}_{R}^{n} (S, S) ≅ (V^{*})^{\otimes n}$ . Let $u, v$ be the dual basis classes in degree one.

2.1step 1.1algebra∎

Yoneda splicing corresponds to concatenating tensors under this resolution. Consequently $u v$ and $v u$ are distinct basis vectors of $(V^{*})^{\otimes 2} = {Ext}_{R}^{2} (S, S)$ . Both products are typed in the same self-Ext algebra, and they differ; this is a genuinely noncommutative Yoneda product.

Depends on

Used by

Nothing in the library uses this result yet.

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