Lemma 37.22.2 (045S)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.22: Cohen-Macaulay morphisms
Lemma 37.22.2 (cite)

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Lemma 37.22.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

$f$ is Cohen-Macaulay, and
$f$ is flat and its fibres are Cohen-Macaulay schemes.

Proof. This follows directly from the definitions. $\square$

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