CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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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.

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Koszul Complex Invariant Under Invertible Generator Change

Statement

If $y_{i} = \sum_{j} a_{i j} x_{j}$ and the matrix $(a_{i j})$ is invertible, the induced generator-matrix chain map is an isomorphism $K (y; M) ≅ K (x; M)$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Generator Matrix Chain Map.

Proof

technique · direct

1.1

The matrix chain map is defined by exterior powers of the generator map.

givenalgebra

2.1

The inverse matrix induces its inverse in every exterior degree, so the complexes are isomorphic.

step 1.1algebra∎

Depends on

Koszul Generator Matrix Chain Map

Used by

Dependency tree · two levels

4 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

The Stacks Project, Koszul complexes and regular sequences (standard reference, not scraped)