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 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.

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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