Semigroups of valuations on local rings, II

TL;DR

This paper establishes general bounds on the growth of valuation ideals in noetherian local domains and explores the geometric shape and complexity of associated valuation semigroups in the value group.

Contribution

It introduces new growth bounds for valuation ideals and analyzes the shape of valuation semigroups in the value group, extending previous results.

Findings

Growth bounds restrict valuation semigroup structures

Shape of valuation semigroups can be highly complex

Examples demonstrate the wild nature of semigroup shapes

Abstract

Given a noetherian local domain $R$ and a valuation $\nu$ of its field of fractions which is non negative on $R$, we derive some very general bounds on the growth of the number of distinct valuation ideals of $R$ corresponding to values lying in certain parts of the value group $\Gamma$ of $\nu$. We show that this growth condition imposes restrictions on the semigroups $\nu(R\setminus \{0\})$ for noetherian $R$ which are stronger that those resulting from the previous paper \cite{C2} of the first author. Given an ordered embedding $\Gamma\subset ({\mathbf R}^h)_{\hbox{\rm lex}}$, where $h$ is the rank of $\nu$, we also study the shape in ${\mathbf R}^h$ of the parts of $\Gamma$ which appear naturally in this study. We give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup $\nu(R\setminus \{0\})$.