Un exemple effectif de gradué non noethérien associé à une valuation divisorielle. (An effective example of a non noetherian graduate ring associated to a divisorial valuation)
type: Item
instance of: scholarly article
title: Un exemple effectif de gradué non noethérien associé à une valuation divisorielle. (An effective example of a non noetherian graduate ring associated to a divisorial valuation)
zbMATH Open document ID: 0966.13001
author: Olivier Piltant
author: Vincent Cossart
author: Carlos Galindo
publication date: 2000-03-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75409
review text: For a local regular ring \(R\) and with the cubic \(x^2z+xy^2+y^3\) the authors construct a sequence of blow-ups \(\text{Spec }R=X_0\leftarrow X_1\leftarrow\dots\leftarrow X_n\), \(n\geq 10\) and a divisorial valuation \(\nu\), so that, for \(I_\alpha=\{ r\in R\mid \nu(r)\geq\alpha\}\), \(\text{gr}_\nu(R)=\bigoplus I_\alpha/I_{\alpha+1}\) is not Noetherian (for \(n\leq 9\), \(\text{gr}_\nu(R)\) is Noetherian). They also prove that the Poincaré series of \(\text{gr}_\nu(R)\) is a rational fraction.
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 13E99
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 1409065
zbMATH Keywords: divisorial valuation
zbMATH Keywords: local regular ring
zbMATH Keywords: not Noetherian graded algebra
zbMATH Keywords: Poincaré series
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
OpenAlex ID: W2324228734
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DOI: 10.5802/AIF.1748
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published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 1409065