Theorem 41.10.1 (0255)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 41: Étale Morphisms of Schemes
Section 41.10: Topological properties of flat morphisms
Theorem 41.10.1 (cite)

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Theorem 41.10.1. Let $Y$ be a locally Noetherian scheme. Let $f : X \to Y$ be a morphism which is locally of finite type. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$ -module. The set of points in $X$ where $\mathcal{F}$ is flat over $Y$ is an open set. In particular the set of points where $f$ is flat is open in $X$ .

Proof. See More on Morphisms, Theorem 37.15.1. $\square$

Comments (2)

Comment #2255 by Almonds on October 18, 2016 at 09:39

$Y$ should be replaced by $S$ , I guess?

Comment #2289 by Johan on November 03, 2016 at 23:13

Thanks. Fixed here.

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