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

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Localization preserves the Cohen--Macaulay depth--dimension equality

Statement

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

Let $(R, m)$ be Noetherian local and let $M$ be a nonzero finite Cohen--Macaulay $R$ -module. For every $p \in {Supp}_{R} (M)$ , ${depth}_{R_{p}} (M_{p}) = \dim_{R_{p}} (M_{p}) .$

Facts & Assumptions

Given: the Axiom of Choice, $M_{p} \neq 0$ ; write $d = \dim_{R} (M)$ .

[L1]

Under Choice, localization gives ${depth}_{R_{p}} (M_{p}) + \dim (R / p) \geq {depth}_{R} (M)$ (Localization gives the stated depth inequality).

[L2]

A nonzero finite local module has depth at most its dimension (A finite local module has depth at most its dimension).

Proof

technique · direct

1.1L1given

By [L1] and Cohen--Macaulayness, ${depth}_{R_{p}} (M_{p}) + \dim (R / p) \geq d .$ The dimension-chain inequality for the support gives $\dim_{R_{p}} (M_{p}) + \dim (R / p) \leq d .$

2.1L2step 1.1algebra∎

Hence localized depth is at least localized dimension. The general bound [L2] supplies the reverse inequality, so equality holds.

Depends on

Used by

Dependency tree · two levels

12 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)