The Stacks project

Proof. The implications (3) $\Rightarrow $ (2) $\Rightarrow $ (1) are immediate.

Let $X \to Y$ be an alteration with $X$ regular. Then $Y^\nu \to Y$ is finite by Lemma 89.7.1. Consider the factorization $f : X \to Y^\nu $ from Morphisms of Spaces, Lemma 67.49.8. The morphism $f$ is finite over an open $V \subset Y^\nu $ containing every point of codimension $\leq 1$ in $Y^\nu $ by Spaces over Fields, Lemma 72.3.2. Then $f$ is flat over $V$ by Algebra, Lemma 10.128.1 and the fact that a normal local ring of dimension $\leq 2$ is Cohen-Macaulay by Serre's criterion (Algebra, Lemma 10.157.4). Then $V$ is regular by Algebra, Lemma 10.164.4. As $Y^\nu $ is Noetherian we conclude that $Y^\nu \setminus V = \{ y_1, \ldots , y_ m\} $ is finite. For each $i$ let $\mathcal{O}_{Y^\nu , y_ i}^ h$ be the henselian local ring. Then $X \times _ Y \mathop{\mathrm{Spec}}(\mathcal{O}_{Y^\nu , y_ i}^ h)$ is a regular alteration of $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y^\nu , y_ i}^ h)$ (some details omitted). By Lemma 89.7.2 the completion of $\mathcal{O}_{Y^\nu , y_ i}^ h$ is normal. In this way we see that (1) $\Rightarrow $ (4).

Assume (4). We have to prove (3). We may immediately replace $Y$ by its normalization. Let $y_1, \ldots , y_ m \in |Y|$ be the singular points. Choose a collection of elementary étale neighbourhoods $(V_ i, v_ i) \to (Y, y_ i)$ as in Section 89.3. For each $i$ the henselian local ring $\mathcal{O}_{Y^\nu , y_ i}^ h$ is the henselization of $\mathcal{O}_{V_ i, v_ i}$. Hence these rings have isomorphic completions. Thus by the result for schemes (Resolution of Surfaces, Theorem 54.14.5) we see that there exist finite sequences of normalized blowups

blowing up only in points lying over $v_ i$ such that $X_{i, n_ i}$ is regular. By Lemma 89.5.3 there is a sequence of normalized blowing ups

and of course $X_ n$ is regular too (look at the local rings). This completes the proof. $\square$