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 54.13.1. Consider the factorization $f : X \to Y^\nu $ from Morphisms, Lemma 29.54.5. The morphism $f$ is finite over an open $V \subset Y^\nu $ containing every point of codimension $\leq 1$ in $Y^\nu $ by Varieties, Lemma 33.17.2. 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. By Lemma 54.13.3 the completion of $\mathcal{O}_{Y^\nu , y_ i}$ 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. Applying Lemmas 54.14.4 and 54.14.3 we find there exists a finite sequence of normalized blowups

such that $Y_{i, n_ i}$ is regular. By Lemma 54.11.7 there is a corresponding sequence of normalized blowing ups

Then $X_{i, n_ i}$ is a regular scheme by Lemma 54.11.2. By Lemma 54.6.5 we can fit these normalized blowing ups into a corresponding sequence

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