LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)

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.

H One Regular Local Implies Koszul Regular

Statement

Assume the Axiom of Choice. For finite $M$ over Noetherian local $(R, m)$ with $x \subseteq m$ , $M$ - $H_{1}$ -regularity implies $M$ -Koszul-regularity.

Facts & Assumptions

Given: The Axiom of Choice (The Axiom of Choice) and the ring, finite module, and sequence stated in the claim. The declared prerequisites used here are Koszul Regular And H One Regular Sequences, Local Koszul H One Detects First Regularity Failure, Local Koszul Acyclicity Iff Regular Sequence, and Assuming the Axiom of Choice, Nakayama's lemma.

Proof

technique · direct

1.1givenalgebra

If $M = 0$ , every term of $K (x; M)$ is zero and the conclusion is immediate. Suppose $M \neq 0$ . Under the assumed AC, Nakayama shows that $M / (x) M \neq 0$ because $(x) \subseteq m$ ; the same argument applies to every prefix quotient. This is the first use of Choice.

2.1step 1.1algebra∎

If $x$ were not regular, it would therefore have a first injectivity failure. The detection lemma would give $H_{1} (K (x; M)) \neq 0$ , contradicting $H_{1}$ -regularity. Thus $x$ is regular, and the AC-qualified local regularity criterion gives vanishing of every positive Koszul homology group. This is the second use of Choice.

Depends on

Used by

Regularity Notions Coincide Local Finite Corollary

Dependency tree · two levels

19 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)