In this paper the question of which semigroups are realizable as the
semigroup of values attained on a Noetherian local ring which is dominated by a
valuation is considered. We give some striking examples, indicating that there
may be no constraints on the semigroup beyond those known classically.Comment: 19 page

Cutkosky, Steven Dale

Teissier, Bernard

English

arXiv

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 $\nu(R\setminus \{0\})$

Hal-Diderot

Semigroups of valuations on local rings, II

Steven Dale Cutkosky

Bernard Teissier

CiteSeerX

SEMIGROUPS OF VALUATIONS ON LOCAL RINGS, II

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.142.7841

Semigroups of valuations on local rings

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Hal-Diderot

CiteSeerX