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 Mapping Cone Homology Exact Sequence

Statement

Appending $y$ yields the exact sequence $\dots \to H_{i} (K (x; M)) \overset{y}{\to} H_{i} (K (x; M)) \to H_{i} (K (x, y; M)) \to H_{i - 1} (K (x; M)) \to \dots$ .

Facts & Assumptions

Given: The rings, finite sequences, modules, and local hypotheses stated in the claim. The declared prerequisites used here are Koszul Append One Element Mapping Cone Identification, The cone long exact sequence.

Proof

technique · direct

1.1givenalgebra

The cone identification gives the degreewise split short exact sequence $0 \to K (x; M) \to K (x, y; M) \to K (x; M) [- 1] \to 0$ .

2.1step 1.1algebra∎

Its connecting map is multiplication by $y$ (check it on a cycle in the shifted summand), so the long exact homology sequence is the displayed one.

Depends on

Dependency tree · 2 levels

Sources

The Stacks Project, Koszul complexes and regular sequences