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

Statement

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 $q$ .

Facts & Assumptions

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.

Proof

technique · direct

1.1

Flat tensoring is exact and therefore commutes with homology quotients.

givenalgebra

2.1

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

step 1.1algebra∎

Depends on

Used by

Functoriality Base Change And Generator Change For Koszul Complexes Theorem

Dependency tree · two levels

6 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)