## 87.8 Resolution

Here is a definition.

Definition 87.8.1. Let $S$ be a scheme. Let $Y$ be a Noetherian integral algebraic space over $S$. A resolution of singularities of $X$ is a modification $f : X \to Y$ such that $X$ is regular.

In the case of surfaces we sometimes want a bit more information.

Definition 87.8.2. Let $S$ be a scheme. Let $Y$ be a $2$-dimensional Noetherian integral algebraic space over $S$. We say $Y$ has a resolution of singularities by normalized blowups if there exists a sequence

$Y_ n \to X_{n - 1} \to \ldots \to Y_1 \to Y_0 \to Y$

where

1. $Y_ i$ is proper over $Y$ for $i = 0, \ldots , n$,

2. $Y_0 \to Y$ is the normalization,

3. $Y_ i \to Y_{i - 1}$ is a normalized blowup for $i = 1, \ldots , n$, and

4. $Y_ n$ is regular.

Observe that condition (1) implies that the normalization $Y_0$ of $Y$ is finite over $Y$ and that the normalizations used in the normalized blowing ups are finite as well. We finally come to the main theorem of this chapter.

Theorem 87.8.3. Let $S$ be a scheme. Let $Y$ be a two dimensional integral Noetherian algebraic space over $S$. 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 $y_1, \ldots , y_ m \in |Y|$ such that the completions of the henselian local rings $\mathcal{O}_{Y^\nu , y_ i}^ h$ are 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 87.7.1. Consider the factorization $f : X \to Y^\nu$ from Morphisms of Spaces, Lemma 65.49.5. 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 70.3.2. 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 10.155.4). Then $V$ is regular by Algebra, Lemma 10.162.4. 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 87.7.2 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 87.3. 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

$X_{i, n_ i} \to X_{i, n_ i - 1} \to \ldots \to V_ i$

blowing up only in points lying over $v_ i$ such that $X_{i, n_ i}$ is regular. By Lemma 87.5.3 there is a sequence of normalized blowing ups

$X_ n \to X_{n - 1} \to \ldots \to X_1 \to Y$

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

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