Lemma 10.163.3 (045J)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.163: Ascending properties
Lemma 10.163.3 (cite)

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Lemma 10.163.3. Let $R \to S$ be a flat local homomorphism of local Noetherian rings. Then the following are equivalent

$S$ is Cohen-Macaulay, and
$R$ and $S/\mathfrak m_ RS$ are Cohen-Macaulay.

Proof. Follows from the definitions and Lemmas 10.163.2 and 10.112.7. $\square$

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