Essential finite generation of valuation rings in characteristic zero algebraic function fields

Abstract

Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega }$ of $\omega$ is essentially finitely generated over the valuation ring $V_{\nu }$ of $\nu$ if and only if the initial index $\epsilon (\omega |\nu )$ is equal to the ramification index $e(\omega |\nu )$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.