LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 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.

The first parameter of a Cohen--Macaulay module is regular

Statement

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

Let $(R, m)$ be Noetherian local and $M$ a nonzero finite Cohen--Macaulay module of positive dimension. If $x_{1}, \dots, x_{d}$ is a system of parameters for $M$ , then $x_{1}$ is $M$ -regular.

Facts & Assumptions

Given: the Axiom of Choice, $d = \dim M > 0$ , and the parameter quotient has dimension $0$ .

[L1]

Under Choice, every associated prime of $M$ has quotient dimension $d$ (Associated primes of a Cohen--Macaulay module have full dimension).

[L2]

Under Choice, an element is a zero divisor on $M$ exactly when it lies in an associated prime; only the union formula is used (Zero divisors on a module over a Noetherian ring are the union of its associated primes).

[L3]

A system of parameters is characterized by its finite-length quotient (For a finite module, the dimension is the least size of an ideal of definition, and such tuples are systems of parameters).

Proof

technique · direct

1.1L1given

If $x_{1}$ lay in an associated prime $p$ of $M$ , then [L1] would give $\dim (R / p) = d$ . Moreover $p \in Supp (M / x_{1} M)$ because $Supp (M / x_{1} M) = Supp (M) \cap V (x_{1})$ .

2.1L2L3step 1.1algebra∎

The remaining $d - 1$ elements make $M / x_{1} M / (x_{2}, \dots, x_{d}) (M / x_{1} M)$ finite length. The parameter characterization [L3] therefore gives $\dim (M / x_{1} M) \leq d - 1$ . This contradicts step 1.1. Thus $x_{1}$ avoids every associated prime; [L2] makes it $M$ -regular. The quotient is nonzero by Nakayama.

Depends on

Used by

Induction along a Cohen--Macaulay parameter sequence Lemma

Dependency tree · two levels

20 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)