The Gorenstein property of the associated graded rings of powers of an ideal
type: Item
instance of: scholarly article
title: The Gorenstein property of the associated graded rings of powers of an ideal
zbMATH Open document ID: 0772.13003
DOI: 10.1016/0022-4049(92)90004-Y
published in: Journal of Pure and Applied Algebra
publication date: 1992-09-27T00:00:00Z
review text: This paper gives a second proof of the following theorem: Let \((R,m)\) be a local Noetherian ring with \(\dim(R)=d\geq 1\) and with infinite residue field. Let \(I\) be an \(m\)-primary ideal of \(R\) such that the associated graded ring \(G(I)\) is Gorenstein. Then for any integer \(r\geq 1\), \(G(I^ r)\) is Gorenstein if and only if \(\delta(I)\equiv d-1\pmod r\), where \(\delta(I)\) is the smallest integer \(i\) such that \(I^{i+1}=JI^ i\) for some parameter ideal \(J\) contained in \(I\). The author had given an earlier proof using canonical modules in J. Algebra 155, No. 2, 397-414 (1993). The proof given in this paper uses Hilbert functions and the fact that under certain conditions \(G(I)\) is Gorenstein if and only if it is Cohen-Macaulay and the Hilbert function is symmetric. The paper ends by determining when \(G(m^ r)\) and \(R(m^ r)\) are Gorenstein for \(R\) a local Gorenstein ring with emb\((R)=e(R)+\dim R-3\).
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13E05
zbMATH DE Number: 64644
zbMATH Keywords: associated graded ring
zbMATH Keywords: Gorenstein
zbMATH Keywords: Hilbert function
author: Akira Ooishi
reviewed by: D. D. Anderson
MaRDI profile type: Publication
cites work: Flawless O-sequences and Hilbert functions of Cohen-Macaulay integral domains
cites work: On the Gorensteinness of Rees algebras over local rings
cites work: Genera and arithmetic genera of commutative rings
cites work: \(\Delta\)-genera and sectional genera of commutative rings
cites work: On the Gorenstein property of the associated graded ring and the Rees algebra of an ideal
Wikidata QID: Q126352114
description: scientific article; zbMATH DE number 64644