When is the Rees algebra Gorenstein?
type: Item
instance of: scholarly article
title: When is the Rees algebra Gorenstein?
zbMATH Open document ID: 0827.13012
author: Trung Ngo Viet
author: Duong Quoc Viet
author: Santiago Zarzuela
published in: Journal of Algebra
publication date: 1995-12-11T00:00:00Z
review text: Let \((A,m)\) be a local ring with \(\dim A \geq 1\) which has a canonical module \(K_A\), \(F\) a sequence of ideals \(I_0 = A \supseteq I_1 \supseteq I_2 \supseteq \ldots\) of \(A\) with \(I_m I_n \subseteq I_{m + n}\) for all \(m,n \geq 0\) and \(R(F) : = \bigoplus_{n \geq 0} I_n\), \(G (F) = \bigoplus_{n \geq 0} I_n/I_{n + 1}\). Then (i) If \(R(F)\) is a Cohen-Macaulay ring then there is a filtration \(\{K_n\}\) of \(K_A\) such that \(K_{R(F)} = \bigoplus_{n \geq 1} K_n\), \(K_{G(F)} = \bigoplus_{n \geq 1} K_{n - 1}/K_n\) are the canonical modules of \(R(F)\), respectively \(G(F)\). (ii) \(R(F)\) is a Gorenstein ring if and only if \(\bigoplus_{n \geq 2} [K_{G(F)}]_n \cong G (F)_{(- 2)}\) and \(R(F)\) is Cohen- Macaulay. By \textit{S. Goto} and \textit{K. Nishida} [Mem. Am. Math. Soc. 526 (1994; Zbl 0812.13016)] and \textit{Duong Quôc Viêt} [Commun. Algebra 21, No. 1, 221-229 (1993; Zbl 0772.13002)], the condition that \(R(F)\) is Cohen- Macaulay is expressed by means of the local cohomology modules of \(G(F)\). Thus (ii) characterizes the property of \(R(F)\) to be Gorenstein in terms of \(G(F)\). These results applied for the \(I\)-adic filtrations, \(I \subseteq A\), yield in particular a criterion of \textit{S. Ikeda} [Nagoya Math. J. 102, 135-154 (1986; Zbl 0585.13014)] valid also for the case grade \((I) = 1\). If grade \((I) = 1\) then there exists a Cohen-Macaulay local ring \(A\) which is not Gorenstein but \(R(I)\) is Gorenstein [there are no such examples when \(\text{grade} (I) \geq 2\) by the cited paper of Ikeda].
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13A30
zbMATH DE Number: 783915
zbMATH Keywords: Rees algebras
zbMATH Keywords: ideal filtrations
zbMATH Keywords: Cohen-Macaulay property
zbMATH Keywords: Gorenstein property
MaRDI profile type: Publication
full work available at URL: jabr.1995.1179
OpenAlex ID: W1993394303
DOI: 10.1006/JABR.1995.1179
description: scientific article; zbMATH DE number 783915