Lemma 76.26.7 (0E10)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.26: Cohen-Macaulay morphisms
Lemma 76.26.7 (cite)

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Lemma 76.26.7. 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 étale 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 37.22.7. $\square$

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