LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedverified 2026-09-24 (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.

Induction along a Cohen--Macaulay parameter sequence

Statement

Assume the Axiom of Choice (The Axiom of Choice).

Under the hypotheses of lem-cohen-macaulay-parameter-first-element-regular, $M / x_{1} M$ is Cohen--Macaulay of dimension $d - 1$ , and $x_{2}, \dots, x_{d}$ is a system of parameters for it.

Facts & Assumptions

Given: The Axiom of Choice; $x_{1}, \dots, x_{d}$ is a system of parameters of the nonzero Cohen--Macaulay module $M$ , with $d = \dim M > 0$ .

Proof

technique · direct

1.1given

The first-element lemma makes $x_{1}$ regular. Exact parameter dimension drop gives $\dim (M / x_{1} M) = d - 1$ , and the regular-quotient equivalence makes the quotient Cohen--Macaulay.

2.1step 1.1algebra∎

Its quotient by $x_{2}, \dots, x_{d}$ is the original parameter quotient, which has dimension $0$ . Since the remaining tuple has length $d - 1$ , it is a system of parameters for $M / x_{1} M$ .

Depends on

Used by

Every system of parameters is regular in a Cohen--Macaulay module Corollary

Dependency tree · two levels

16 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

Depth and Cohen--Macaulay modules source treatment (standard reference, not scraped)