Definition 89.8.1. 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.
89.8 Resolution
Here is a definition.
In the case of surfaces we sometimes want a bit more information.
Definition 89.8.2. 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.
\[ Y_ n \to X_{n - 1} \to \ldots \to Y_1 \to Y_0 \to Y \]
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 89.8.3. 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, $Y^\nu $ has finitely many singular points $y_1, \ldots , y_ m \in |Y|$, and for each $i$ the completion of the henselian local ring $\mathcal{O}_{Y^\nu , y_ i}^ h$ 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 89.7.1. Consider the factorization $f : X \to Y^\nu $ from Morphisms of Spaces, Lemma 67.49.8. 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 72.3.2. 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. 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 89.7.2 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 89.3. 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 89.5.3 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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