Lemma 29.53.15 (03GR)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.53: Relative normalization
Lemma 29.53.15 (cite)

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Lemma 29.53.15. Let $f : X \to S$ be a morphism. Assume that

$S$ is a Nagata scheme,
$f$ is of finite type,
$X$ is reduced.

Then the normalization $\nu : S' \to S$ of $S$ in $X$ is finite.

Proof. This is a special case of Lemma 29.53.14. Namely, (2) holds as the finite type morphism $f$ is quasi-compact by definition and quasi-separated by Lemma 29.15.7. Condition (3) holds because $X$ is locally Noetherian by Lemma 29.15.6. Finally, condition (4) holds because a finite type morphism induces finitely generated residue field extensions. $\square$

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