Depth properties of Rees algebras and associated graded rings
type: Item
instance of: scholarly article
title: Depth properties of Rees algebras and associated graded rings
zbMATH Open document ID: 0813.13010
author: Sam Huckaba
author: Thomas Marley
published in: Journal of Algebra
publication date: 1995-05-30T00:00:00Z
review text: Let \(R\) be a Cohen-Macaulay local ring of dimension \(\geq 1\) with maximal ideal \(M\) and \(I\) an \(M\)-primary ideal. This paper is interested in determining the depth of the associated graded ring \(gr_ I(R)\) and in connecting \(\text{depth} gr_ I(R)\) with the depth of the Rees algebra \(R[It]\). The main result of this paper is that if \(gr_ I(R)\) is not Cohen-Macaulay, then \(\text{depth} R[It] = \text{depth} gr_ I(R) + 1\). Moreover, if \(R\) is a regular local ring, then this equality holds even if \(gr_ I (R)\) is Cohen-Macaulay. Several results are also given about when the reduction number of an ideal is independent of the choice of minimal reduction.
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13C14
zbMATH DE Number: 495550
zbMATH Keywords: Cohen-Macaulay local ring
zbMATH Keywords: depth of the associated graded ring
zbMATH Keywords: Rees algebra
reviewed by: David F. Anderson
MaRDI profile type: Publication
full work available at URL: jabr.1993.1075
OpenAlex ID: W2021343379
DOI: 10.1006/JABR.1993.1075
description: scientific article; zbMATH DE number 495550