The Stacks project

[Theorem on page 151, Lipman]

Theorem 54.14.5 (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, $Y^\nu $ has finitely many singular points $y_1, \ldots , y_ m$, and for each $y_ i$ 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 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

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

such that $Y_{i, n_ i}$ is regular. By Lemma 54.11.7 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 54.11.2. By Lemma 54.6.5 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 (2)

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).

Comment #4757 by on

Thanks very much. I have a hard time catching things like this myself. See changes here.

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0BGP. Beware of the difference between the letter 'O' and the digit '0'.