The Stacks project

Apologies if this is already somewhere in the Stacks project, but the cited Lemma 10.163.3 also shows the following.

Let $f \colon X \rightarrow Y$ be a Cohen–Macaulay morphism of locally Noetherian schemes. If $x \in X$ is such that $\mathcal{O}_{Y, f(x) }$ is Cohen–Macaulay,

then $\mathcal{O}_{X,x}$ is also Cohen–Macaulay.

And sorry for this, but: I wonder if it's better to use "Cohen–Macaulay" instead of "Cohen-Macaulay", i.e. an n-dash instead of a hyphen.

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The weird formatting above is because the comment box resued to let me typeset the two local rings on the same line...