#### Vol. 16, No. 2, 2022

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

### Steven Dale Cutkosky

Vol. 16 (2022), No. 2, 291–310
##### 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 $𝜖\left(\omega |\nu \right)$ is equal to the ramification index $e\left(\omega |\nu \right)$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.

##### Keywords
valuation ring, initial index, ramification, essential finite generation
Primary: 13A18
Secondary: 14E15
##### Milestones
Received: 5 December 2019
Revised: 19 January 2021
Accepted: 24 February 2021
Published: 27 April 2022
##### Authors
 Steven Dale Cutkosky Department of Mathematics University of Missouri Columbia, MO United States