The Stacks project

Lemma 74.26.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is flat and locally of finite presentation. Let

\[ W = \{ x \in |X| : f\text{ is Cohen-Macaulay at }x\} \]

Then $W$ is open in $|X|$ and the formation of $W$ commutes with arbitrary base change of $f$: For any morphism $g : Y' \to Y$, consider the base change $f' : X' \to Y'$ of $f$ and the projection $g' : X' \to X$. Then the corresponding set $W'$ for the morphism $f'$ is equal to $W' = (g')^{-1}(W)$.

Proof. Choose a commutative diagram

\[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

with ├ętale vertical arrows and $U$ and $V$ schemes. Let $u \in U$ with image $x \in |X|$. Then $f$ is Cohen-Macaulay at $x$ if and only if $U \to V$ is Cohen-Macaulay at $u$ (by definition). Thus we reduce to the case of the morphism $U \to V$. See More on Morphisms, Lemma 37.20.7. $\square$


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