Generating sequences of valuations on simple extensions of domains

TL;DR

This paper characterizes the structure of the associated graded ring of polynomial extensions over valuation rings in simple domain extensions, providing explicit descriptions under certain conditions and connecting to MacLane's key polynomials.

Contribution

It offers a complete description of the associated graded ring in simple extensions when there is a unique valuation extension and residue characteristic conditions are met, using MacLane's key polynomials.

Findings

Explicit description of the associated graded ring in simple extensions

Construction of key polynomials within the domain's polynomial ring

Extension of previous results to more general residue field conditions

Abstract

Suppose that $(K,v_0)$ is a valued field, $f(x)\in K[x]$ is a monic and irreducible polynomial and $(L,v)$ is an extension of valued fields, where $L=K[x]/(f(x))$. Let $A$ be a local domain with quotient field $K$ dominated by the valuation ring of $v_0$ and such that $f(x)$ is in $A[x]$. The study of these extensions is a classical subject. This paper is devoted to the problem of describing the structure of the associated graded ring ${\rm gr}_v A[x]/(f(x))$ of $A[x]/(f(x))$ for the filtration defined by $v$ as an extension of the associated graded ring of $A$ for the filtration defined by $v_0$. We give a complete simple description of this algebra when there is unique extension of $v_0$ to $L$ and the residue characteristic of $A$ does not divide the degree of $f$. To do this, we show that the sequence of key polynomials constructed by MacLane's algorithm can be taken to lie inside $A[x]$. This result was proven using a different method in the more restrictive case that the residue fields of $A$ and of the valuation ring of $v$ are equal and algebraically closed in a recent paper by Cutkosky, Mourtada and Teissier.