Lemma 38.11.2 (081P)—The Stacks project

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Part 2: Schemes
Chapter 38: More on Flatness
Section 38.11: Extending properties from an open
Lemma 38.11.2 (cite)

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Lemma 38.11.2. Let $f : X \to S$ be a morphism of schemes. Let $U \subset S$ be open. Assume

$f$ is locally of finite type and flat,
$U \subset S$ is retrocompact and scheme theoretically dense,
$f|_{f^{-1}U} : f^{-1}U \to U$ is locally of finite presentation.

Then $f$ is of locally of finite presentation.

Proof. The question is local on $X$ and $S$, hence we may assume $X$ and $S$ affine. Choose a closed immersion $i : X \to \mathbf{A}^ n_ S$ and apply Lemma 38.11.1 to $i_*\mathcal{O}_ X$. Some details omitted. $\square$

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