Lemma 33.27.3 (0BXS)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 33: Varieties
Section 33.27: Normalization
Lemma 33.27.3 (cite)

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Lemma 33.27.3. Let $k$ be a field. Let $f : Y \to X$ be a quasi-compact morphism of locally algebraic schemes over $k$ . Let $X'$ be the normalization of $X$ in $Y$ . If $Y$ is reduced, then $X' \to X$ is finite.

Proof. Since $Y$ is quasi-separated (by Properties, Lemma 28.5.4 and Morphisms, Lemma 29.15.6) the morphism $f$ is quasi-separated (Schemes, Lemma 26.21.13). Hence Morphisms, Definition 29.53.3 applies. The result follows from Morphisms, Lemma 29.53.14. This uses that locally algebraic schemes are locally Noetherian (hence have locally finitely many irreducible components) and that locally algebraic schemes are Nagata (Morphisms, Lemma 29.18.2). Some small details omitted. $\square$

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