Lemma 54.8.7. Let A be a Nagata regular local ring of dimension 2. Then A defines a rational singularity.
Proof. (The assumption that A be Nagata is not necessary for this proof, but we've only defined the notion of rational singularity in the case of Nagata 2-dimensional normal local domains.) Let X \to \mathop{\mathrm{Spec}}(A) be a modification with X normal. By Lemma 54.4.2 we can dominate X by a scheme X_ n which is the last in a sequence
of blowing ups in closed points. By Lemma 54.3.2 the schemes X_ i are regular, in particular normal (Algebra, Lemma 10.157.5). By Lemma 54.8.1 we have H^1(X, \mathcal{O}_ X) \subset H^1(X_ n, \mathcal{O}_{X_ n}). Thus it suffices to prove H^1(X_ n, \mathcal{O}_{X_ n}) = 0. Using Lemma 54.8.1 again, we see that it suffices to prove R^1(X_ i \to X_{i - 1})_*\mathcal{O}_{X_ i} = 0 for i = 1, \ldots , n. This follows from Lemma 54.3.4. \square
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