The Stacks project

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. $\square$

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