MacLane–Vaquié chains of valuations on a polynomial ring

Enric Nart

Vol. 311 (2021), No. 1, 165–195

DOI: 10.2140/pjm.2021.311.165

Abstract

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

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