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
is open in the fibre X_ s.
Proof. Suppose x \in U. Choose an elementary étale 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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