Essential Finite Generation of Valuation Rings in Characteristic Zero Algebraic Function Fields

TL;DR

This paper proves that in characteristic zero algebraic function fields, the valuation ring extension is essentially finitely generated if and only if the initial index equals the ramification index, answering a question by Knaf.

Contribution

It establishes a precise criterion linking initial and ramification indices for finite extensions of valuation rings in characteristic zero.

Findings

Valuation ring $V_{\omega}$ is essentially finitely generated over $V_{ u}$ if and only if $\epsilon(\omega| u) = e(\omega| u)$.

Provides a positive answer to Knaf's question in characteristic zero.

Clarifies the structure of valuation ring extensions in 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.