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.

One Element Koszul Homology

Statement

For $K (x; M)$ , $H_{0} = M / x M$ , $H_{1} = (0 :_{M} x)$ , and $H_{i} = 0$ for $i \notin {0, 1}$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are One Element Koszul Complex, Homology object of a chain complex.

Proof

technique · direct

1.1

The sole differential is multiplication by $x$ , whose cokernel is $M / x M$ .

givenalgebra

2.1

Its kernel is $(0 :_{M} x)$ and no other chain groups occur.

step 1.1algebra∎

Depends on

Used by

Empty And Unit Koszul Boundaries Example
Koszul Homology Of A Zero Divisor Example
Koszul Homology Zero Divisor Example
Regular One Element Koszul Acyclicity Lemma
Basic Koszul Homology 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)