The Stacks project

Lemma 87.7.1. Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over $S$. Assume there exists an alteration $f : X \to Y$ with $X$ regular. Then the normalization $Y^\nu \to Y$ is finite and $Y$ has a dense open which is regular.

Proof. By ├ętale localization, it suffices to prove this when $Y = \mathop{\mathrm{Spec}}(A)$ where $A$ is a Noetherian domain. Let $B$ be the integral closure of $A$ in its fraction field. Set $C = \Gamma (X, \mathcal{O}_ X)$. By Cohomology of Spaces, Lemma 67.20.2 we see that $C$ is a finite $A$-module. As $X$ is normal (Properties of Spaces, Lemma 64.25.4) we see that $C$ is normal domain (Spaces over Fields, Lemma 70.4.6). Thus $B \subset C$ and we conclude that $B$ is finite over $A$ as $A$ is Noetherian.

There exists a nonempty open $V \subset Y$ such that $f^{-1}V \to V$ is finite, see Spaces over Fields, Definition 70.8.3. After shrinking $V$ we may assume that $f^{-1}V \to V$ is flat (Morphisms of Spaces, Proposition 65.32.1). Thus $f^{-1}V \to V$ is faithfully flat. Then $V$ is regular by Algebra, Lemma 10.164.4. $\square$

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