Lemma 10.103.5 (0C6G)—The Stacks project

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Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.103: Cohen-Macaulay modules
Lemma 10.103.5 (cite)

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Lemma 10.103.5. Let $R$ be a Noetherian local ring with maximal ideal $\mathfrak m$ . Let $M$ be a finite $R$ -module. Let $x \in \mathfrak m$ be a nonzerodivisor on $M$ . Then $M$ is Cohen-Macaulay if and only if $M/xM$ is Cohen-Macaulay.

Proof. By Lemma 10.72.7 we have $\text{depth}(M/xM) = \text{depth}(M)-1$ . By Lemma 10.63.10 we have $\dim (\text{Supp}(M/xM)) = \dim (\text{Supp}(M)) - 1$ . $\square$

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