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.

One Element Koszul Complex

Statement

Let $R$ be a commutative unital ring, let $M$ be an $R$ -module, and let $x \in R$ . Then $K (x; M)$ is $0 \to M \overset{x}{\to} M \to 0$ , with the left copy in degree $1$ .

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.

Proof

technique · direct

1.1

For one basis element, the only exterior powers are $R e$ and $R$ .

givenalgebra

2.1

The defining differential sends $e \otimes m$ to $x m$ , producing the displayed two-term complex.

step 1.1algebra∎

Depends on

Koszul Complex Of A Sequence With Coefficients

Used by

One Element Koszul Homology Corollary
Koszul Complex One And Two Elements Example

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)