Three facts of valuation theory
type: Item
instance of: scholarly article
title: Three facts of valuation theory
zbMATH Open document ID: 0928.13001
author: Bernard Teissier
published in: Proceedings of the Japan Academy. Series A
publication date: 1999-05-03T00:00:00Z
review text: The author gives a sketch of results and their proofs that are to appear in the Obergurgl conference proceedings, 1997. For a field \(k\), a local integral \(k\)-algebra \((R,m)\) with a field of fractions \(K\), this is a study of \(k\)-valuations \(v\) of \(K\) with center in \(R\) (their valuation ring \((R_v, m_v)\) contains \(R\) and \(m_v\cap k=(0)\)). The results concern the following: (1) The Spivakovsky graded algebra is isomorphic to a certain quotient of a polynomial ring in possibly countably many indeterminates. Thus it is a deformation of a (non-normal) toric variety, possibly of infinite embedding dimension. (2) The graded \(k\)\(_v\)-algebra, \(\text{gr}_vR_v\), is a filtering direct limit of termic maps between polynomial subalgebras in rank \(r(\Phi)\) variables (\(\Phi\) is the totally ordered group of the valuation). (3) Let \(R\) be an analytically irreducible Noetherian local ring, \(v_1\) the valuation of height one with which \(v\) is composed, \(p=m_{v_1}\cap R\), the center of \(v_1\) on \(R\); then the completion \(\widetilde R^v\) with respect to the topology defined by the ideal filtration \(P_i\), \(i\in\Phi_+\), is isomorphic (as topological rings) to a quotient of the \(p\)-adic completion \(\widetilde R^p\) of \(R\). Moreover, it is Noetherian and in particular, if \(R\) is excellent, so is \(\widetilde R^v\).
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 13J10
zbMATH DE Number: 1268559
zbMATH Keywords: valuations
zbMATH Keywords: polynomial subalgebras
zbMATH Keywords: graded rings
zbMATH Keywords: toric geometry
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: pjaa.74.107
OpenAlex ID: W2095083631
cites work: Q5687941
cites work: Binomial ideals
cites work: Classification of \(A\)-graded algebras with 3 generators
cites work: Q4052252
cites work: Komplettierung semilokaler quasiausgezeichneter Ringe
cites work: Valuations in Function Fields of Surfaces
cites work: Q4078152
cites work: Q4440914
cites work: Local uniformization on algebraic varieties
cites work: Q5732721
DOI: 10.3792/PJAA.74.107
description: scientific article; zbMATH DE number 1268559