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.

## 87.8 Resolution

Here is a definition.

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

where

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

$Y_0 \to Y$ is the normalization,

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

$Y_ n$ 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

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

there exists a resolution of singularities of $Y$,

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

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

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

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

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