MacLane-Vaqui\'e chains of valuations on a polynomial ring

TL;DR

This paper reviews MacLane and Vaquié's results on extending valuations from a field to its polynomial ring, introducing chains of valuations that describe all such extensions and their algebraic structures.

Contribution

It introduces MacLane-Vaquié chains of valuations on polynomial rings and proves their fundamental role in classifying all valuations extending a given valuation.

Findings

Every valuation on $K[x]$ is a limit of a MacLane-Vaquié chain.

The chains are essentially unique and contain explicit algebraic data.

The graded algebra of a valuation can be described over the base valuation's graded algebra.

Abstract

Let $(K,v)$ be a valued field. We review some results of MacLane and Vaqui\'e on extensions of $v$ to valuations on the polynomial ring $K[x]$. We introduce certain MacLane-Vaqui\'e chains of residually transcendental valuations, and we prove that every valuation $\mu$ on $K[x]$ is a limit of a finite or countably infinite MacLane-Vaqui\'e chain. This chain underlying $\mu$ is essentially unique and contains arithmetic data yielding an explicit description of the graded algebra of $\mu$ as an algebra over the graded algebra of $v$.