CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 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 Homology Flat Base Change

For flat $R \to A$ , $H_{q} (K_{R} (x; M)) \otimes_{R} A ≅ H_{q} (K_{A} (x A; M \otimes_{R} A))$ for every $q$ .

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Complex Flat Base Change, Flat and faithfully flat modules and ring homomorphisms.

technique · direct

1.1givenalgebra

Flat tensoring is exact and therefore commutes with homology quotients.

2.1step 1.1algebra∎

The base-changed complex is the Koszul complex on $x A$ with coefficient $M \otimes_{R} A$ .

Depends on