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

Statement

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

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, Flat and faithfully flat modules and ring homomorphisms.

Proof

technique · direct

1.1

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.

givenalgebra

2.1

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.

step 1.1algebra∎

Depends on

Used by

Koszul Homology Flat Base Change Corollary
Functoriality Base Change And Generator Change For Koszul Complexes Theorem

Dependency tree · two levels

8 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)