Lemma 37.16.8 (0CZR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 37: More on Morphisms
Section 37.16: Critère de platitude par fibres
Lemma 37.16.8 (cite)

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Lemma 37.16.8. Let $f : X \to S$ be a morphism of schemes which is locally of finite presentation. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}_ X$ -module flat over $S$ . Then the set

$\{ x \in X : \mathcal{F}\text{ free in a neighbourhood of }x\}$

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

Proof. Openness holds trivially. Let $x \in X$ mapping to $s \in S$ . By Lemma 37.16.7 we see that $x$ is in our set if and only if $\mathcal{F}|_{X_ s}$ is flat at $x$ over $X_ s$ . Clearly this is also equivalent to $\mathcal{F}$ being flat at $x$ over $X$ (because this statement is implied by freeness of $\mathcal{F}_ x$ and implies flatness of $\mathcal{F}|_{X_ s}$ at $x$ over $X_ s$ ). Thus the base change statement follows from Lemma 37.16.5 applied to $\text{id} : X \to X$ over $S$ . $\square$

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