The Stacks project

Lemma 10.120.18. Let $A$ be a Noetherian domain of dimension $1$ with fraction field $K$. Let $L/K$ be a finite extension. Let $B$ be the integral closure of $A$ in $L$. Then $B$ is a Dedekind domain and $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is surjective, has finite fibres, and induces finite residue field extensions.

Proof. By Krull-Akizuki (Lemma 10.119.12) the ring $B$ is Noetherian. By Lemma 10.112.4 $\dim (B) = 1$. Thus $B$ is a Dedekind domain by Lemma 10.120.17. Surjectivity of the map on spectra follows from Lemma 10.36.17. The last two statements follow from Lemma 10.119.10. $\square$

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