The Stacks project

Proof. By étale 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 69.20.2 we see that $C$ is a finite $A$-module. As $X$ is normal (Properties of Spaces, Lemma 66.25.4) we see that $C$ is normal domain (Spaces over Fields, Lemma 72.4.6). 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 72.8.3. After shrinking $V$ we may assume that $f^{-1}V \to V$ is flat (Morphisms of Spaces, Proposition 67.32.1). Thus $f^{-1}V \to V$ is faithfully flat. Then $V$ is regular by Algebra, Lemma 10.164.4. $\square$

Proof. Parts (1) and (2) follow from Lemma 89.7.1. We have to redo part of the proof of that lemma in order to set up notation for the proof of (3). Set $C = \Gamma (X, \mathcal{O}_ X)$. By Cohomology of Spaces, Lemma 69.20.2 we see that $C$ is a finite $A$-module. As $X$ is normal (Properties of Spaces, Lemma 66.25.4) we see that $C$ is normal domain (Spaces over Fields, Lemma 72.4.6). Thus $B \subset C$ and we conclude that $B$ is finite over $A$ as $A$ is Noetherian. By Resolution of Surfaces, Lemma 54.13.2 in order to prove (3) it suffices to show that the $\mathfrak m$-adic completion $C^\wedge $ is normal.

By Algebra, Lemma 10.97.8 the completion $C^\wedge $ is the product of the completions of $C$ at the prime ideals of $C$ lying over $\mathfrak m$. There are finitely many of these and these are the maximal ideals $\mathfrak m_1, \ldots , \mathfrak m_ r$ of $C$. (The corresponding result for $B$ explains the final statement of the lemma.) Thus replacing $A$ by $C_{\mathfrak m_ i}$ and $X$ by $X_ i = X \times _{\mathop{\mathrm{Spec}}(C)} \mathop{\mathrm{Spec}}(C_{\mathfrak m_ i})$ we reduce to the case discussed in the next paragraph. (Note that $\Gamma (X_ i, \mathcal{O}) = C_{\mathfrak m_ i}$ by Cohomology of Spaces, Lemma 69.11.2.)

Here $A$ is a Noetherian local normal domain and $f : X \to \mathop{\mathrm{Spec}}(A)$ is a regular alteration with $\Gamma (X, \mathcal{O}_ X) = A$. We have to show that the completion $A^\wedge $ of $A$ is a normal domain. By Lemma 89.6.2 $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ is regular. Since $\Gamma (Y, \mathcal{O}_ Y) = A^\wedge $ by Cohomology of Spaces, Lemma 69.11.2. We conclude that $A^\wedge $ is normal as before. Namely, $Y$ is normal by Properties of Spaces, Lemma 66.25.4. It is connected because $\Gamma (Y, \mathcal{O}_ Y) = A^\wedge $ is local. Hence $Y$ is normal and integral (as connected and normal implies integral for separated algebraic spaces). Thus $\Gamma (Y, \mathcal{O}_ Y) = A^\wedge $ is a normal domain by Spaces over Fields, Lemma 72.4.6. This proves (3).

Proof of (4). Let $\eta \in \mathop{\mathrm{Spec}}(A)$ denote the generic point and denote by a subscript $\eta $ the base change to $\eta $. Since $f$ is an alteration, the scheme $X_\eta $ is finite and faithfully flat over $\eta $. Since $Y = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A^\wedge )$ is regular by Lemma 89.6.2 we see that $Y_\eta $ is regular (as a limit of opens in $Y$). Then $Y_\eta \to \mathop{\mathrm{Spec}}(A^\wedge \otimes _ A \kappa (\eta ))$ is finite faithfully flat onto the generic formal fibre. We conclude by Algebra, Lemma 10.164.4. $\square$