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

Cutkosky, Steven Dale

doi:10.2140/ant.2022.16.291

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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

DOI: 10.2140/ant.2022.16.291

Abstract

Let $K$ be a characteristic zero algebraic function field with a valuation $ν$ . Let $L$ be a finite extension of $K$ and $ω$ be an extension of $ν$ to $L$ . We establish that the valuation ring $V_{ω}$ of $ω$ is essentially finitely generated over the valuation ring $V_{ν}$ of $ν$ if and only if the initial index $𝜖 (ω | ν)$ is equal to the ramification index $e (ω | ν)$ 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

Mathematical Subject Classification

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

Vol. 16, No. 2, 2022