CorollaryStatement: 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.

Sequence Ideal Annihilates Koszul Homology

Statement

Every $x_{i}$ , hence the ideal $(x)$ , annihilates every $H_{q} (K (x; M))$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Generator Contraction Homotopy.

Proof

technique · direct

1.1

For each generator, the contraction identity is $d h_{i} + h_{i} d = x_{i} id$ .

givenalgebra

2.1

Thus each $x_{i}$ acts trivially on homology, hence so does every element of $(x)$ .

step 1.1algebra∎

Depends on

Koszul Generator Contraction Homotopy

Used by

Koszul Homology Supported On Sequence Vanishing Set Corollary
Koszul Generators Act Null Homotopically Lemma

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)