Lemma 38.13.8 (05IM)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 38: More on Flatness
Section 38.13: Flat finite type modules, Part II
Lemma 38.13.8 (cite)

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Lemma 38.13.8. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Assume the set of weakly associated points of $S$ is locally finite in $S$ . Then the set of points $x \in X$ where $f$ is flat is an open subscheme $U \subset X$ and $U \to S$ is flat and locally of finite presentation.

Proof. The problem is local on $X$ and $S$ , hence we may assume that $X$ and $S$ are affine. Then $X \to S$ corresponds to a finite type ring map $A \to B$ . Choose a surjection $A[x_1, \ldots , x_ n] \to B$ and consider $B$ as an $A[x_1, \ldots , x_ n]$ -module. An application of Lemma 38.13.7 finishes the proof. $\square$

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