The Stacks project

Lemma 33.27.1. Let $k$ be a field. Let $X$ be a locally algebraic scheme over $k$. Let $\nu : X^\nu \to X$ be the normalization morphism, see Morphisms, Definition 29.54.1. Then

  1. $\nu $ is finite, dominant, and $X^\nu $ is a disjoint union of normal irreducible locally algebraic schemes over $k$,

  2. $\nu $ factors as $X^\nu \to X_{red} \to X$ and the first morphism is the normalization morphism of $X_{red}$,

  3. if $X$ is a reduced algebraic scheme, then $\nu $ is birational,

  4. if $X$ is a variety, then $X^\nu $ is a variety and $\nu $ is a finite birational morphism of varieties.

Proof. Since $X$ is locally of finite type over a field, we see that $X$ is locally Noetherian (Morphisms, Lemma 29.15.6) hence every quasi-compact open has finitely many irreducible components (Properties, Lemma 28.5.7). Thus Morphisms, Definition 29.54.1 applies. The normalization $X^\nu $ is always a disjoint union of normal integral schemes and the normalization morphism $\nu $ is always dominant, see Morphisms, Lemma 29.54.5. Since $X$ is universally Nagata (Morphisms, Lemma 29.18.2) we see that $\nu $ is finite (Morphisms, Lemma 29.54.11). Hence $X^\nu $ is locally algebraic too. At this point we have proved (1).

Part (2) is Morphisms, Lemma 29.54.2.

Part (3) is Morphisms, Lemma 29.54.7.

Part (4) follows from (1), (2), (3), and the fact that $X^\nu $ is separated as a scheme finite over a separated scheme. $\square$

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