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

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

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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