Lemma 67.49.9 (0BB5)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.49: Normalization
Lemma 67.49.9 (cite)

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Lemma 67.49.9. Let $S$ be a scheme. Let $X$ be a Nagata algebraic space over $S$ . The normalization $\nu : X^\nu \to X$ is a finite morphism.

Proof. Since $X$ being Nagata is locally Noetherian, Definition 67.49.6 applies. By construction of $X^\nu$ in Lemma 67.49.5 we immediately reduce to the case of schemes which is Morphisms, Lemma 29.54.11. $\square$

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