Lemma 77.4.2. 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 étale morphism $V \to Y$ mapping $v$ to $y$. Choose a scheme $U$ and a surjective étale 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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