The Stacks project

Lemma 10.103.6. Let $R \to S$ be a surjective homomorphism of Noetherian local rings. Let $N$ be a finite $S$-module. Then $N$ is Cohen-Macaulay as an $S$-module if and only if $N$ is Cohen-Macaulay as an $R$-module.

Proof. Omitted. $\square$

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