Lemma 76.26.5. Let $S$ be a scheme. Let $f : X \to Y$ be a flat morphism of locally Noetherian algebraic spaces over $S$. If $X$ is Cohen-Macaulay, then $f$ is Cohen-Macaulay and $\mathcal{O}_{Y, f(\overline{x})}$ is Cohen-Macaulay for all $x \in |X|$.
Proof. After translating into algebra using Lemma 76.26.3 (compare with the proof of Lemma 76.26.4) this follows from Algebra, Lemma 10.163.3. $\square$
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