A Valuation Theorem for Noetherian Rings

Abstract

Let $\mathcal{A}\subset \mathcal{B}$ be integral domains. Suppose $\mathcal{A}$ is Noetherian and $\mathcal{B}$ is a finitely generated $\mathcal{A}$-algebra. Denote by $\bar{\mathcal{A}}$ the integral closure of $\mathcal{A}$ in $\mathcal{B}$. We show that $\bar{\mathcal{A}}$ is determined by finitely many unique discrete valuation rings. Our result generalizes Rees’ classical valuation theorem for ideals. We also obtain a variant of Zariski’s main theorem.