Lemma 37.20.5. Let $f : X \to Y$ be a flat morphism of locally Noetherian schemes. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\mathcal{O}_{Y, f(x)}$ is Cohen-Macaulay for all $x \in X$.

Proof. After translating into algebra this follows from Algebra, Lemma 10.161.3. $\square$

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