Loading web-font TeX/Math/Italic

The Stacks project

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


Comments (0)


Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.