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 Complex Flat Base Change

For a flat map $R \to A$ , there is a natural chain isomorphism $K_{R} (x; M) \otimes_{R} A ≅ K_{A} (x A; M \otimes_{R} A)$ .

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, Flat and faithfully flat modules and ring homomorphisms.

technique · direct

1.1givenalgebra

In degree $p$ , base change sends $(e_{I} \otimes m) \otimes a$ to $e_{I} \otimes (m \otimes a)$ ; the exterior basis makes this an isomorphism.

2.1step 1.1algebra∎

The two differentials agree on every basis wedge by $x_{i} (m) \otimes a = (x_{i} a) (m \otimes 1)$ , which proves the chain claim.

Depends on