Lemma 37.16.4 (039E)—The Stacks project

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Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.16: Critère de platitude par fibres
Lemma 37.16.4 (cite)

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Lemma 37.16.4. 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 $S$,
for every $s \in S$ the morphism $f_ s : X_ s \to Y_ s$ is flat, and
$Y$ is locally of finite type over $S$.

Then $f$ is flat. If $f$ is also surjective, then $Y$ is flat over $S$.

Proof. This is a special case of Theorem 37.16.2. $\square$

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