Lemma 76.23.6 (05X3)—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 76.23.6 (cite)

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Lemma 76.23.6. 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 $Z$ , and
$Y$ is locally of finite type over $Z$ .

Then the set

$\{ x \in |X| : X\text{ flat at }x \text{ over }Y\} .$

is open in $|X|$ and its formation commutes with arbitrary base change $Z' \to Z$ .

Proof. This is a special case of Lemma 76.23.5. $\square$

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