The Stacks project

Lemma 38.10.4. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module of finite type. Let $s \in S$. Then the set

\[ \{ x \in X_ s \mid \mathcal{F} \text{ flat over }S\text{ at }x\} \]

is open in the fibre $X_ s$.

Proof. Suppose $x \in U$. Choose an elementary ├ętale neighbourhood $(S', s') \to (S, s)$ and open $V \subset X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S', s'})$ as in Proposition 38.10.3. Note that $X_{s'} = X_ s$ as $\kappa (s) = \kappa (s')$. If $x' \in V \cap X_{s'}$, then the pullback of $\mathcal{F}$ to $X \times _ S S'$ is flat over $S'$ at $x'$. Hence $\mathcal{F}$ is flat at $x'$ over $S$, see Morphisms, Lemma 29.25.13. In other words $X_ s \cap V \subset U$ is an open neighbourhood of $x$ in $U$. $\square$

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