MacLane–Vaquié chains of valuations on a polynomial ring

Let $(K, v)$ be a valued field. We reinterpret some results of MacLane and Vaquié on extensions of $v$ to valuations on the polynomial ring $K [x]$ . We introduce certain MacLane–Vaquié chains constructed as a mixture of ordinary and limit augmentations of valuations. Every valuation $ν$ on $K [x]$ is a limit (in a certain sense) of a countable MacLane–Vaquié chain. This chain underlying $ν$ is essentially unique and contains discrete arithmetic data yielding an explicit description of the graded algebra of $ν$ as an algebra over the graded algebra of $v$ .

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Keywords

graded algebra, key polynomial, MacLane–Vaquié chain, valuation

Mathematical Subject Classification

Primary: 13A18

Secondary: 12J20

Milestones

Received: 14 September 2020

Revised: 18 December 2020

Accepted: 8 January 2021

Published: 17 March 2021

Authors

Enric Nart
Departament de Matemàtiques Universitat Autònoma de Barcelona Cerdanyola del Vallès Catalonia Spain

Vol. 311, No. 1, 2021

Enric Nart

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification

Milestones

Authors