Valuations dominating regular local rings and proximity relations
type: Item
instance of: scholarly article
title: Valuations dominating regular local rings and proximity relations
zbMATH Open document ID: 1114.13017
published in: Journal of Pure and Applied Algebra
publication date: 2007-02-26T00:00:00Z
review text: Let \(R\) be a (commutative) regular noetherian local ring of dimension \(n\geq 2\) and let \(V\) be a valuation ring of the quotient field of \(R\) such that \(V\) dominates \(R\). There is a sequence of regular noetherian local rings (of dimension \(n\)) associated with \((R,V)\), \((R_i): R_0\subset R_1\subset\dots\subset R_i\subset\dots\subset V\), such that \(R_i\) is the quadratic transform of \(R_{i-1}\) along \(V\), for \(i\geq 1\). The main purpose of this paper is to study the proximity relations for valuations centered in such regular local rings. One question is when \(V=\bigcup_{i=0}^\infty R_i\). If \(v\) is the valuation associated with \(V\) that is zero-dimensional, a related question is: what information about \(v\) is recoverable from the sequence \((R_i)\) and vice-versa. The case \(n=2\) being fairly settled, the authors introduce the proximity matrix and the multiplicity sequence (thus extending the classical definitions for the case \(n=2\)), associated with the sequence \((R_i)\). The authors describe precise relations between these invariants and look into their properties.
reviewed by: Radoslav M. Dimitrić
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13H05
zbMATH DE Number: 5129947
author: Ángel Granja
author: Ma. Carmen Martínez
author: Cristina Rodríguez
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.06.010
OpenAlex ID: W1968324920
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DOI: 10.1016/J.JPAA.2006.06.010
description: scientific article; zbMATH DE number 5129947