54.14 Resolution

Here is a definition.

Definition 54.14.1. Let $Y$ be a Noetherian integral scheme. A resolution of singularities of $Y$ 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 54.14.2. Let $Y$ be a $2$-dimensional Noetherian integral scheme. We say $Y$ has a resolution of singularities by normalized blowups if there exists a sequence

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

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.

Lemma 54.14.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Assume $A$ is normal and has dimension $2$. If $\mathop{\mathrm{Spec}}(A)$ has a resolution of singularities, then $\mathop{\mathrm{Spec}}(A)$ has a resolution by normalized blowups.

Proof. By Lemma 54.13.3 the completion $A^\wedge$ of $A$ is normal. By Lemma 54.11.2 we see that $\mathop{\mathrm{Spec}}(A^\wedge )$ has a resolution. By Lemma 54.11.7 any sequence $Y_ n \to Y_{n - 1} \to \ldots \to \mathop{\mathrm{Spec}}(A^\wedge )$ of normalized blowups of comes from a sequence of normalized blowups $X_ n \to \ldots \to \mathop{\mathrm{Spec}}(A)$. Moreover if $Y_ n$ is regular, then $X_ n$ is regular by Lemma 54.11.2. Thus it suffices to prove the lemma in case $A$ is complete.

Assume in addition $A$ is a complete. We will use that $A$ is Nagata (Algebra, Proposition 10.162.16), excellent (More on Algebra, Proposition 15.52.3), and has a dualizing complex (Dualizing Complexes, Lemma 47.22.4). Moreover, the same is true for any ring essentially of finite type over $A$. If $B$ is a excellent local normal domain, then the completion $B^\wedge$ is normal (as $B \to B^\wedge$ is regular and More on Algebra, Lemma 15.42.2 applies). We will use this without further mention in the rest of the proof.

Let $X \to \mathop{\mathrm{Spec}}(A)$ be a resolution of singularities. Choose a sequence of normalized blowing ups

$Y_ n \to Y_{n - 1} \to \ldots \to Y_1 \to \mathop{\mathrm{Spec}}(A)$

dominating $X$ (Lemma 54.5.3). The morphism $Y_ n \to X$ is an isomorphism away from finitely many points of $X$. Hence we can apply Lemma 54.4.2 to find a sequence of blowing ups

$X_ m \to X_{m - 1} \to \ldots \to X$

in closed points such that $X_ m$ dominates $Y_ n$. Diagram

$\xymatrix{ & Y_ n \ar[rd] \ar[rr] & & \mathop{\mathrm{Spec}}(A) \\ X_ m \ar[rr] \ar[ru] & & X \ar[ru] }$

To prove the lemma it suffices to show that a finite number of normalized blowups of $Y_ n$ produce a regular scheme. By our diagram above we see that $Y_ n$ has a resolution (namely $X_ m$). As $Y_ n$ is a normal surface this implies that $Y_ n$ has at most finitely many singularities $y_1, \ldots , y_ t$ (because $X_ m \to Y_ n$ is an isomorphism away from the fibres of dimension $1$, see Varieties, Lemma 33.17.3).

Let $x_ a \in X$ be the image of $y_ a$. Then $\mathcal{O}_{X, x_ a}$ is regular and hence defines a rational singularity (Lemma 54.8.7). Apply Lemma 54.8.4 to $\mathcal{O}_{X, x_ a} \to \mathcal{O}_{Y_ n, y_ a}$ to see that $\mathcal{O}_{Y_ n, y_ a}$ defines a rational singularity. By Lemma 54.9.8 there exists a finite sequence of blowups in singular closed points

$Y_{a, n_ a} \to Y_{a, n_ a - 1} \to \ldots \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y_ n, y_ a})$

such that $Y_{a, n_ a}$ is Gorenstein, i.e., has an invertible dualizing module. By (the essentially trivial) Lemma 54.6.4 with $n' = \sum n_ a$ these sequences correspond to a sequence of blowups

$Y_{n + n'} \to Y_{n + n' - 1} \to \ldots \to Y_ n$

such that $Y_{n + n'}$ is normal and the local rings of $Y_{n + n'}$ are Gorenstein. Using the references given above we can dominate $Y_{n + n'}$ by a sequence of blowups $X_{m + m'} \to \ldots \to X_ m$ dominating $Y_{n + n'}$ as in the following

$\xymatrix{ & Y_{n + n'} \ar[rr] & & Y_ n \ar[rd] \ar[rr] & & \mathop{\mathrm{Spec}}(A) \\ X_{m + m'} \ar[ru] \ar[rr] & & X_ m \ar[rr] \ar[ru] & & X \ar[ru] }$

Thus again $Y_{n + n'}$ has a finite number of singular points $y'_1, \ldots , y'_ s$, but this time the singularities are rational double points, more precisely, the local rings $\mathcal{O}_{Y_{n + n'}, y'_ b}$ are as in Lemma 54.12.3. Arguing exactly as above we conclude that the lemma is true. $\square$

Lemma 54.14.4. Let $(A, \mathfrak m, \kappa )$ be a Noetherian complete local ring. Assume $A$ is a normal domain of dimension $2$. Then $\mathop{\mathrm{Spec}}(A)$ has a resolution of singularities.

Proof. A Noetherian complete local ring is J-2 (More on Algebra, Proposition 15.48.7), Nagata (Algebra, Proposition 10.162.16), excellent (More on Algebra, Proposition 15.52.3), and has a dualizing complex (Dualizing Complexes, Lemma 47.22.4). Moreover, the same is true for any ring essentially of finite type over $A$. If $B$ is a excellent local normal domain, then the completion $B^\wedge$ is normal (as $B \to B^\wedge$ is regular and More on Algebra, Lemma 15.42.2 applies). In other words, the local rings which we encounter in the rest of the proof will have the required “excellency” properties required of them.

Choose $A_0 \subset A$ with $A_0$ a regular complete local ring and $A_0 \to A$ finite, see Algebra, Lemma 10.160.11. This induces a finite extension of fraction fields $K/K_0$. We will argue by induction on $[K : K_0]$. The base case is when the degree is $1$ in which case $A_0 = A$ and the result is true.

Suppose there is an intermediate field $K_0 \subset L \subset K$, $K_0 \not= L \not= K$. Let $B \subset A$ be the integral closure of $A_0$ in $L$. By induction we choose a resolution of singularities $Y \to \mathop{\mathrm{Spec}}(B)$. Let $X$ be the normalization of $Y \times _{\mathop{\mathrm{Spec}}(B)} \mathop{\mathrm{Spec}}(A)$. Picture:

$\xymatrix{ X \ar[r] \ar[d] & \mathop{\mathrm{Spec}}(A) \ar[d] \\ Y \ar[r] & \mathop{\mathrm{Spec}}(B) }$

Since $A$ is J-2 the regular locus of $X$ is open. Since $X$ is a normal surface we conclude that $X$ has at worst finitely many singular points $x_1, \ldots , x_ n$ which are closed points with $\dim (\mathcal{O}_{X, x_ i}) = 2$. For each $i$ let $y_ i \in Y$ be the image. Since $\mathcal{O}_{Y, y_ i}^\wedge \to \mathcal{O}_{X, x_ i}^\wedge$ is finite of smaller degree than before we conclude by induction hypothesis that $\mathcal{O}_{X, x_ i}^\wedge$ has resolution of singularities. By Lemma 54.14.3 there is a sequence

$Z^\wedge _{i, n_ i} \to \ldots \to Z^\wedge _{i, 1} \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i}^\wedge )$

of normalized blowups with $Z^\wedge _{i, n_ i}$ regular. By Lemma 54.11.7 there is a corresponding sequence of normalized blowing ups

$Z_{i, n_ i} \to \ldots \to Z_{i, 1} \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i})$

Then $Z_{i, n_ i}$ 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

$Z_ n \to Z_{n - 1} \to \ldots \to Z_1 \to X$

and of course $Z_ n$ is regular too (look at the local rings). This proves the induction step.

Assume there is no intermediate field $K_0 \subset L \subset K$ with $K_0 \not= L \not= K$. Then either $K/K_0$ is separable or the characteristic to $K$ is $p$ and $[K : K_0] = p$. Then either Lemma 54.8.6 or 54.8.10 implies that reduction to rational singularities is possible. By Lemma 54.8.5 we conclude that there exists a normal modification $X \to \mathop{\mathrm{Spec}}(A)$ such that for every singular point $x$ of $X$ the local ring $\mathcal{O}_{X, x}$ defines a rational singularity. Since $A$ is J-2 we find that $X$ has finitely many singular points $x_1, \ldots , x_ n$. By Lemma 54.9.8 there exists a finite sequence of blowups in singular closed points

$X_{i, n_ i} \to X_{i, n_ i - 1} \to \ldots \to \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x_ i})$

such that $X_{i, n_ i}$ is Gorenstein, i.e., has an invertible dualizing module. By (the essentially trivial) Lemma 54.6.4 with $n = \sum n_ a$ these sequences correspond to a sequence of blowups

$X_ n \to X_{n - 1} \to \ldots \to X$

such that $X_ n$ is normal and the local rings of $X_ n$ are Gorenstein. Again $X_ n$ has a finite number of singular points $x'_1, \ldots , x'_ s$, but this time the singularities are rational double points, more precisely, the local rings $\mathcal{O}_{X_ n, x'_ i}$ are as in Lemma 54.12.3. Arguing exactly as above we conclude that the lemma is true. $\square$

We finally come to the main theorem of this chapter.

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. $\square$

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