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The Stacks project

Lemma 54.9.8. Let (A, \mathfrak m, \kappa ) be a local normal Nagata domain of dimension 2 which defines a rational singularity. Assume A has a dualizing complex. Then there exists a finite sequence of blowups in singular closed points

X = X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = \mathop{\mathrm{Spec}}(A)

such that X_ i is normal for each i and such that the dualizing sheaf \omega _ X of X is an invertible \mathcal{O}_ X-module.

Proof. The dualizing module \omega _ A is a finite A-module whose stalk at the generic point is invertible. Namely, \omega _ A \otimes _ A K is a dualizing module for the fraction field K of A, hence has rank 1. Thus there exists a blowup b : Y \to \mathop{\mathrm{Spec}}(A) such that the strict transform of \omega _ A with respect to b is an invertible \mathcal{O}_ Y-module, see Divisors, Lemma 31.35.3. By Lemma 54.5.3 we can choose a sequence of normalized blowups

X_ n \to X_{n - 1} \to \ldots \to X_1 \to \mathop{\mathrm{Spec}}(A)

such that X_ n dominates Y. By Lemma 54.9.4 and arguing by induction each X_ i \to X_{i - 1} is simply a blowing up.

We claim that \omega _{X_ n} is invertible. Since \omega _{X_ n} is a coherent \mathcal{O}_{X_ n}-module, it suffices to see its stalks are invertible modules. If x \in X_ n is a regular point, then this is clear from the fact that regular schemes are Gorenstein (Dualizing Complexes, Lemma 47.21.3). If x is a singular point of X_ n, then each of the images x_ i \in X_ i of x is a singular point (because the blowup of a regular point is regular by Lemma 54.3.2). Consider the canonical map f_ n^*\omega _ A \to \omega _{X_ n} of Lemma 54.9.6. For each i the morphism X_{i + 1} \to X_ i is either a blowup of x_ i or an isomorphism at x_ i. Since x_ i is always a singular point, it follows from Lemma 54.9.7 and induction that the maps f_ i^*\omega _ A \to \omega _{X_ i} is always surjective on stalks at x_ i. Hence

(f_ n^*\omega _ A)_ x \longrightarrow \omega _{X_ n, x}

is surjective. On the other hand, by our choice of b the quotient of f_ n^*\omega _ A by its torsion submodule is an invertible module \mathcal{L}. Moreover, the dualizing module is torsion free (Duality for Schemes, Lemma 48.22.3). It follows that \mathcal{L}_ x \cong \omega _{X_ n, x} and the proof is complete. \square


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