LemmaStatement: Literature-sourcedProof: 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.

Koszul Generator Matrix Chain Map

Statement

If $y_{i} = \sum_{j} a_{i j} x_{j}$ , the transpose matrix defines an exterior map inducing a chain map $K (y; M) \to K (x; M)$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Of A Sequence With Coefficients, Koszul Differential Coordinate Formula, Chain map.

Proof

technique · direct

1.1givenalgebra

Let $E_{y} \to E_{x}$ send the $i$ th basis vector to $\sum_{j} a_{i j} e_{j}$ . The assumed identity $y_{i} = \sum_{j} a_{i j} x_{j}$ makes the degree-one squares commute.

2.1step 1.1algebra∎

Extending by exterior powers respects wedge products, and the derivation rule then makes the resulting map commute with the differentials in every degree.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Koszul complexes and regular sequences