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.

Regular Sequence Koszul Acyclicity Induction

Statement

If $H_{i} (K (x; M)) = 0$ for every $i > 0$ and multiplication by $y$ is injective on $M / (x) M$ , then all positive homology of $K (x, y; M)$ vanishes.

Facts & Assumptions

Given: The ring, finite sequence, module, and element stated in the claim. The declared prerequisites used here are Basic Koszul Homology and Koszul Mapping Cone Homology Exact Sequence.

Proof

technique · direct

1.1givenalgebra

Put $C = K (x; M)$ . By hypothesis $H_{i} (C) = 0$ for $i > 0$ , while $H_{0} (C) = M / (x) M$ by the basic Koszul-homology calculation.

2.1step 1.1algebra∎

The mapping-cone exact sequence for $K (x, y; M)$ identifies its $H_{1}$ with the kernel of multiplication by $y$ on $H_{0} (C)$ , because $H_{1} (C) = 0$ ; this kernel is zero by hypothesis. For $i > 1$ , the adjacent groups $H_{i} (C)$ and $H_{i - 1} (C)$ both vanish, so exactness gives $H_{i} (K (x, y; M)) = 0$ .

Depends on

Used by

Regular Sequences Give Acyclic 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)