Lemma 76.23.4 (05X1)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 76: More on Morphisms of Spaces
Section 76.23: Critère de platitude par fibres
Lemma 76.23.4 (cite)

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Lemma 76.23.4. Let $S$ be a scheme. Let $f : X \to Y$ and $Y \to Z$ be a morphism of algebraic spaces over $S$ . Assume

$X$ is locally of finite presentation over $Z$ ,
$X$ is flat over $Z$ ,
for every $z \in |Z|$ the fibre of $X$ over $z$ is flat over the fibre of $Y$ over $z$ , and
$Y$ is locally of finite type over $Z$ .

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

Proof. This is a special case of Theorem 76.23.3. $\square$

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