On the Gorenstein property of form rings
type: Item
instance of: scholarly article
title: On the Gorenstein property of form rings
zbMATH Open document ID: 0797.13001
DOI: 10.1007/BF03025723
author: Jürgen Ribbe
author: Manfred Herrmann
author: Peter Schenzel
published in: Mathematische Zeitschrift
publication date: 1994-10-13T00:00:00Z
full work available at URL: https://eudml.org/doc/174531
review text: This paper deals with the Gorenstein property of the Rees algebra \(R(I) = \bigoplus_{n \geq 0} I^ nt^ n\) and the form ring \(G(I) = \bigoplus_{n \geq 0} I^ n/I^{n+1}\) of an ideal \(I\) in a local Gorenstein ring \((A,{\mathfrak m})\). Here are the main results of the paper for an ideal \(I\) of height \(\geq 2\): \(\bullet\) The following two conditions are equivalent: (1) \(R(I^ r)\) is Gorenstein and \(R(I)\) is Cohen-Macaulay. (2) \(G(I)\) is Gorenstein with \(a(G(I)) = - (r+1)\), where \(a=a(G(I))\) is the \(a\)-invariant of \(G(I)\). \(\bullet\) The following three conditions are equivalent: (1) \(G(I)\) is a Gorenstein ring. (2) \(G(I^ r)\) is a Gorenstein ring for all \(r| a+1\). (3) \(G(I^ r)\) is a Gorenstein ring for some \(r \geq 1\). Examples are provided to delimit the scope of the presented results.
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 549773
zbMATH Keywords: canonical module
zbMATH Keywords: Gorenstein property of the Rees algebra
zbMATH Keywords: form ring
zbMATH Keywords: Gorenstein ring
reviewed by: Moshe Roitman
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 549773