CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passverified 2026-09-24 (gpt-6-sol)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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One regular system of parameters implies Cohen--Macaulayness

Statement

Assume the Axiom of Choice.

Let $0 \neq M$ be finite over a Noetherian local ring. If one system of parameters for $M$ is $M$ -regular, then $M$ is Cohen--Macaulay. Here a system of parameters for $M$ means a tuple $x_{1}, \dots, x_{d}$ in the maximal ideal, where $d = \dim {Supp}_{R} (M)$ , such that $M / (x_{1}, \dots, x_{d}) M$ has finite length.

Facts & Assumptions

Given: The Axiom of Choice, a nonzero finite module $M$ over a Noetherian local ring and an $M$ -regular parameter tuple of length $d = \dim {Supp}_{R} (M)$ , using the terminology introduced in 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

1.1given

The regular parameter system gives ${depth}_{R} (M) \geq d$ . Under the stated AC, Depth is bounded by support dimension gives ${depth}_{R} (M) \leq d$ .

2.1step 1.1algebra∎

Hence depth and dimension both equal $d$ , which is Cohen--Macaulayness.

Depends on

Used by

Dependency tree · two levels

15 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)