The Stacks project

Lemma 67.48.11. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume that

  1. $Y$ is Nagata,

  2. $f$ is quasi-separated of finite type,

  3. $X$ is reduced.

Then the normalization $\nu : Y' \to Y$ of $Y$ in $X$ is finite.

Proof. The question is ├ętale local on $Y$, see Lemma 67.48.4. Thus we may assume $Y = \mathop{\mathrm{Spec}}(R)$ is affine. Then $R$ is a Noetherian Nagata ring and we have to show that the integral closure of $R$ in $\Gamma (X, \mathcal{O}_ X)$ is finite over $R$. Since $f$ is quasi-compact we see that $X$ is quasi-compact. Choose an affine scheme $U$ and a surjective ├ętale morphism $U \to X$ (Properties of Spaces, Lemma 66.6.3). Then $\Gamma (X, \mathcal{O}_ X) \subset \Gamma (U, \mathcal{O}_ X)$. Since $R$ is Noetherian it suffices to show that the integral closure of $R$ in $\Gamma (U, \mathcal{O}_ U)$ is finite over $R$. As $U \to Y$ is of finite type this follows from Morphisms, Lemma 29.53.15. $\square$

Comments (2)

Comment #5093 by Tongmu He on

Typo: the conclusion of 67.48.11 seems reversing the roles of X and Y.

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