# Dedekind Domains and Affine Varieties

Let $k$ be an algebraically closed field, and let $B$ be a finitely generated $k$ algebra that is also a Domain. Then $B$ is the affine coordinate ring of some affine variety $Y$; this part is straight out of Hartshorne and is not terribly difficult to understand. However, I am having trouble making sense of the following extension of this statement.

If $B$ is Dedekind, then $Y$ is both non-singular and has dimension 1.

This shows up in Hartshorne (page 41, Lemma 6.5, last paragraph).

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Let $P$ be a non-zero prime ideal of $B$. Since $P$ is a maximal ideal, ht($P$) = 1. Hence dim $B$ = 1.
It remains to prove that $B_P$ is regular. Since $B_P$ is Noetherian, integrally closed and dim $B_P = 1$, $B_P$ is a discrete valuation ring(e.g. Atiyah-MacDonald). Hence $B_P$ is regular.
These are essentially part of the definition. Namely, one can define a Dedekind domain as an integrally closed, noetherian, dimension one domain. There is an important result that says that since $B$ is normal, the singular points of $Y=\operatorname{Spec}B$ have codimension $\ge 2.$ Since $\dim Y=1,$ the subvariety $\operatorname{Sing}Y=\emptyset$ must be empty.