Lemma 37.20.2. Let $f : X \to Y$ be a morphism of schemes. Assume all fibres of $f$ are locally Noetherian. The following are equivalent

1. $f$ is Cohen-Macaulay, and

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

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

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