The Stacks project

[Theorem on page 151, Lipman]

Theorem 53.14.5 (Lipman). Let $Y$ be a two dimensional integral Noetherian scheme. The following are equivalent

  1. there exists an alteration $X \to Y$ with $X$ regular,

  2. there exists a resolution of singularities of $Y$,

  3. $Y$ has a resolution of singularities by normalized blowups,

  4. the normalization $Y^\nu \to Y$ is finite and $Y^\nu $ has finitely many singular points $y_1, \ldots , y_ m$ such that the completion of $\mathcal{O}_{Y^\nu , y_ i}$ is normal.

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 53.13.1. Consider the factorization $f : X \to Y^\nu $ from Morphisms, Lemma 28.52.5. 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 32.17.2. Then $f$ is flat over $V$ by Algebra, Lemma 10.127.1 and the fact that a normal local ring of dimension $\leq 2$ is Cohen-Macaulay by Serre's criterion (Algebra, Lemma 10.152.4). Then $V$ is regular by Algebra, Lemma 10.159.4. As $Y^\nu $ is Noetherian we conclude that $Y^\nu \setminus V = \{ y_1, \ldots , y_ m\} $ is finite. By Lemma 53.13.3 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 53.14.4 and 53.14.3 we find there exists a finite sequence of normalized blowups

\[ Y_{i, n_ i} \to Y_{i, n_ i - 1} \to \ldots \to \mathop{\mathrm{Spec}}(\mathcal{O}^\wedge _{Y, y_ i}) \]

such that $Y_{i, n_ i}$ is regular. By Lemma 53.11.7 there is a corresponding sequence of normalized blowing ups

\[ X_{i, n_ i} \to \ldots \to X_{i, 1} \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y_ i}) \]

Then $X_{i, n_ i}$ is a regular scheme by Lemma 53.11.2. By Lemma 53.6.5 we can fit these normalized blowing ups into a corresponding sequence

\[ X_ n \to X_{n - 1} \to \ldots \to X_1 \to Y \]

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

Comments (1)

Comment #4566 by comment_bot on

In (4), the condition on the is not entirely unambiguous. I suggest rephrasing it to "the normalization is finite, has finitely many singular points , and for each the completion of the local ring is normal." (I am guessing that this is the intended meaning).

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