The Stacks project

Lemma 76.4.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $y \in |Y|$ and $F = f^{-1}(\{ y\} ) \subset |X|$. Then the set

\[ \{ x \in F \mid \mathcal{F} \text{ flat over }Y\text{ at }x\} \]

is open in $F$.

Proof. Choose a scheme $V$, a point $v \in V$, and an ├ętale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$ and a surjective ├ętale morphism $U \to V \times _ Y X$. Then $|U_ v| \to F$ is an open continuous map of topological spaces as $|U| \to |X|$ is continuous and open. Hence the result follows from the case of schemes which is More on Flatness, Lemma 38.10.4. $\square$

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