The Stacks project

Lemma 32.27.2. 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 27.5.4 and Morphisms, Lemma 28.14.6) the morphism $f$ is quasi-separated (Schemes, Lemma 25.21.13). Hence Morphisms, Definition 28.51.3 applies. The result follows from Morphisms, Lemma 28.51.14. 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 28.17.2). Some small details omitted. $\square$


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