Lemma 76.22.2 (05WW)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.22: Openness of the flat locus
Lemma 76.22.2 (cite)

previous theorem

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Lemma 76.22.2. Let $S$ be a scheme. Let

$\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }$

be a cartesian diagram of algebraic spaces over $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Assume $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation. Then

$\{ x' \in |X'| : (g')^*\mathcal{F}\text{ is flat over }Y'\text{ at }x'\}$

is the inverse image of the open subset of Theorem 76.22.1 under the continuous map $|g'| : |X'| \to |X|$ .

Proof. This follows from Morphisms of Spaces, Lemma 67.31.3. $\square$

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