The Stacks project

Lemma 33.27.2. Let $k$ be a field. Let $X$ be a proper scheme over $k$. Let $\nu : X^\nu \to X$ be the normalization morphism, see Morphisms, Definition 29.54.1. Then $X^\nu $ is proper over $k$. If $X$ is projective over $k$, then $X^\nu $ is projective over $k$.

Proof. By Lemma 33.27.1 the morphism $\nu $ is finite. Hence $X^\nu $ is proper over $k$ by Morphisms, Lemmas 29.44.11 and 29.41.4. The morphism $\nu $ is projective by Morphisms, Lemma 29.44.16 and hence if $X$ is projective over $k$, then $X^\nu $ is projective over $k$ by Morphisms, Lemma 29.43.14. $\square$

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