On finite generation of Rees rings defined by filtrations of ideals
type: Item
instance of: scholarly article
title: On finite generation of Rees rings defined by filtrations of ideals
zbMATH Open document ID: 0814.13003
author: Kazuhiko Kurano
published in: Journal of Mathematics of Kyoto University
publication date: 1995-06-12T00:00:00Z
review text: Let \(S\) be a Noetherian ring and \(T = S[V]\) the polynomial ring over \(S\) with the variable \(V\). The author gives sufficient conditions for finite generation of the Rees ring \(R({\mathcal F})\) associated with a filtration \({\mathcal F} = \{F_ i\}_{i \in \mathbb{Z}}\) of ideals of \(T\), when the Rees ring \(R({\mathcal G})\) is Noetherian. Here \({\mathcal G} = \{G_ i\}_{i \in \mathbb{Z}}\) is a filtration of \(S\) such that \(G_ i = F_ i \cap S\). Moreover, the author proves that to investigate whether \(R({\mathcal F})\) is Noetherian or not \(({\mathcal F}\) is a filtration in a base ring \(A)\) we may assume that \(A\) is local. Finally, he investigates the Noetherian property of the symbolic Rees algebra of the ideal defining the ordinary Rees ring associated with an ideal \(I\) of \(A\) such that \(\text{grade} (I) \geq \rho_ A(I) - 1\), where \(\rho_ A (I)\) is the minimal number of generators of \(I\).
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 633574
zbMATH Keywords: finite generation of a Rees ring
zbMATH Keywords: filtration
zbMATH Keywords: Noetherian property
zbMATH Keywords: symbolic Rees algebra
reviewed by: Jiří Močkoř
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519063
OpenAlex ID: W1590507302
DOI: 10.1215/KJM/1250519063
description: scientific article; zbMATH DE number 633574