Lemma 37.16.6 (05VK)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.16: Critère de platitude par fibres
Lemma 37.16.6 (cite)

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Lemma 37.16.6. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of schemes over $S$. Assume

$X$ is locally of finite presentation over $S$,
$X$ is flat over $S$, and
$Y$ is locally of finite type over $S$.

Then the set

\[ U = \{ x \in X \mid X\text{ flat at }x \text{ over }Y\} . \]

is open in $X$ and its formation commutes with arbitrary base change.

Proof. This is a special case of Lemma 37.16.5. $\square$

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