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 Generator Contraction Homotopy

Statement

For each $i$ , exterior multiplication by $e_{i}$ is a degree- $1$ homotopy satisfying $d (e_{i} \land -) + (e_{i} \land -) d = x_{i} id$ on $K (x; M)$ .

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, Exterior Multiplication Koszul Sign Rule, A chain homotopy.

Proof

technique · direct

1.1givenalgebra

Let $h_{i} (z) = e_{i} \land z$ . The graded Leibniz rule gives $d h_{i} (z) + h_{i} d (z) = d (e_{i}) z = x_{i} z$ .

2.1step 1.1algebra∎

Thus $h_{i}$ is a chain homotopy from multiplication by $x_{i}$ to zero, with no division or characteristic assumption.

Depends on

Used by

Sequence Ideal Annihilates Koszul Homology Corollary
Koszul Generators Act Null Homotopically Lemma
shifted adic koszul filtration euler comparison Lemma

Dependency tree · two levels

10 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)