Theorem 37.15.1. Let $S$ be a scheme. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module which is locally of finite presentation. Then

is open in $X$.

[IV Theorem 11.3.1, EGA]

Theorem 37.15.1. Let $S$ be a scheme. Let $f : X \to S$ be a morphism which is locally of finite presentation. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module which is locally of finite presentation. Then

\[ U = \{ x \in X \mid \mathcal{F}\text{ is flat over }S\text{ at }x\} \]

is open in $X$.

**Proof.**
We may test for openness locally on $X$ hence we may assume that $f$ is a morphism of affine schemes. In this case the theorem is exactly Algebra, Theorem 10.128.4.
$\square$

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## Comments (2)

Comment #2691 by Johan on

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