On the graded algebra relative to a valuation
type: Item
instance of: scholarly article
title: On the graded algebra relative to a valuation
zbMATH Open document ID: 0881.13001
DOI: 10.1007/BF02678188
author: Antonio Campillo
author: Carlos Galindo
published in: Manuscripta Mathematica
publication date: 1997-07-07T00:00:00Z
full work available at URL: https://eudml.org/doc/156259
review text: Let \(v\) be a valuation of a field \(F\) which is the quotient field of a noetherian local domain \(R\) centered at \(R\) having residue field \(K\). It is shown that the value semigroup \(S\) of \(v\) is a CF (= combinatorial finite) semigroup. It is also shown how the semigroup theoretical properties of \(S\) allow to describe a minimal system of homogeneous generators of the \(S\)-graded \(K\)-algebra \(\text{gr}_vR\) relative to \(v\) and to obtain an \(S\)-graded minimal resolution of \(\text{gr}_vR\) as \(A[v]\)-module, \(A[v]\) being a certain associated polynomial ring. The paper generalizes some results proved earlier for semigroup \(K\)-algebras [see, e.g., \textit{A. Campillo} and \textit{C. Marijuan} in: Sémin. Théor. Nombres Bordx., Sér. II 3, No. 2, 249-260 (1991; Zbl 0818.20078)].
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13A02
zbMATH DE Number: 1019707
zbMATH Keywords: valuation
zbMATH Keywords: value semigroup
reviewed by: Sudesh Kaur Khanduja
MaRDI profile type: Publication
cites work: Local uniformization on algebraic surfaces over ground fields of characteristic \(p\neq 0\)
cites work: On the Valuations Centered in a Local Domain
cites work: Higher order relations for a numerical semigroup
cites work: Plane valuations and their completions
cites work: The defining ideal of a plane valuation
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Q3739243
cites work: Sandwiched singularities and desingularization of surfaces by normalized Nash transformations
cites work: Valuations in Function Fields of Surfaces
cites work: Non-existence of the Artin function for Henselian pairs
cites work: Q5732721
description: scientific article; zbMATH DE number 1019707