TheoremStatement: 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.

Basic Koszul Homology

Statement

For a finite sequence $x$ , $H_{0} (K (x; M)) = M / (x) M$ , $H_{i} = 0$ for $i > n$ , and $H_{n} = {m \in M : x_{i} m = 0 for all i}$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are One Element Koszul Homology, Koszul Differential Coordinate Formula.

Proof

technique · direct

1.1

The image in degree zero is $(x_{1} M + \dots + x_{n} M)$ , so $H_{0} = M / (x) M$ .

givenalgebra

2.1

There are no degrees above $n$ , and the top differential kills $m$ exactly when every $x_{i}$ kills $m$ .

step 1.1algebra∎

Depends on

Used by

Koszul Complex Resolves A Regular Quotient Corollary
Koszul Complex One And Two Elements Example
Regular Sequence Koszul Acyclicity Induction Lemma

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)