Lemma 37.16.5. Let S be a scheme. Let f : X \to Y be a morphism of schemes over S. Let \mathcal{F} be a quasi-coherent \mathcal{O}_ X-module. Assume
X is locally of finite presentation over S,
\mathcal{F} an \mathcal{O}_ X-module of finite presentation,
\mathcal{F} is flat over S, and
Y is locally of finite type over S.
Then the set
U = \{ x \in X \mid \mathcal{F} \text{ flat at }x \text{ over }Y\} .
is open in X and its formation commutes with arbitrary base change: If S' \to S is a morphism of schemes, and U' is the set of points of X' = X \times _ S S' where \mathcal{F}' = \mathcal{F} \times _ S S' is flat over Y' = Y \times _ S S', then U' = U \times _ S S'.
Proof.
By Morphisms, Lemma 29.21.11 the morphism f is locally of finite presentation. Hence U is open by Theorem 37.15.1. Because we have assumed that \mathcal{F} is flat over S we see that Theorem 37.16.2 implies
U = \{ x \in X \mid \mathcal{F}_ s \text{ flat at }x \text{ over }Y_ s\} .
where s always denotes the image of x in S. (This description also works trivially when \mathcal{F}_ x = 0.) Moreover, the assumptions of the lemma remain in force for the morphism f' : X' \to Y' and the sheaf \mathcal{F}'. Hence U' has a similar description. In other words, it suffices to prove that given s' \in S' mapping to s \in S we have
\{ x' \in X'_{s'} \mid \mathcal{F}'_{s'} \text{ flat at }x' \text{ over }Y'_{s'}\}
is the inverse image of the corresponding locus in X_ s. This is true by Lemma 37.15.2 because in the cartesian diagram
\xymatrix{ X'_{s'} \ar[d] \ar[r] & X_ s \ar[d] \\ Y'_{s'} \ar[r] & Y_ s }
the horizontal morphisms are flat as they are base changes by the flat morphism \mathop{\mathrm{Spec}}(\kappa (s')) \to \mathop{\mathrm{Spec}}(\kappa (s)).
\square
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