The Stacks project

Lemma 51.16.7. Let $(A, \mathfrak m)$ be a Noetherian local ring. Let $I \subset A$ be an ideal. Assume $A$ is excellent, normal, and $\dim V(I) \geq 1$. Then $\text{cd}(A, I) < \dim (A)$. In particular, if $\dim (A) = 2$, then $\mathop{\mathrm{Spec}}(A) \setminus V(I)$ is affine.

Proof. By More on Algebra, Lemma 15.52.6 the completion $A^\wedge $ is normal and hence a domain. Thus the assumption of Proposition 51.16.6 holds and we conclude. The statement on affineness follows from Lemma 51.4.8. $\square$

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