On the Buchsbaum property of associated graded rings
type: Item
instance of: scholarly article
title: On the Buchsbaum property of associated graded rings
zbMATH Open document ID: 0942.13002
author: Yukio Nakamura
published in: Journal of Algebra
publication date: 2000-08-21T00:00:00Z
review text: Let \(I\) be an \({\mathfrak m}\)-primary ideal in a Buchsbaum local ring \((A,{\mathfrak m})\). In this article, the author studies the Buchsbaum property of the associated graded ring \(G(I)=\bigoplus I^n/I^{n+1}\) of \(I\) when the equality \(I^2={\mathfrak q}I\) holds for some minimal reduction \({\mathfrak q}\) of \(I\). Namely: Theorem 1.1. Let \((A,{\mathfrak m})\) be a \(d\)-dimensional Buchsbaum local ring with infinite residue field and \(I\) an \({\mathfrak m}\)-primary ideal such that \(I^2={\mathfrak q}I\) for some parameter ideal \({\mathfrak q}=(a_1,a_2, \dots,a_d)A\) contained in \(I\). Let \(a_i^*\in G(I)\) be the initial form of \(a_i\). Then the following conditions are equivalent: (1) \(G(I)_M\) is a Buchsbaum ring. (2) \(a^*_1,a_2^*, \dots,a^*_d\) form a \(d^+\)-sequence on \(G(I)\). (3) \((a^2_1,a_2^2, \dots, a_d^2)A\cap I^n= (a^2_1,a^2_2, \dots, a^2_d)I^{n-2}\) for all \(3\leq n\leq d+1\). Corollary 1.2. Let \((A,{\mathfrak m})\) be a \(d\)-dimensional Buchsbaum local ring with infinite residue field and \(I\) an \({\mathfrak m}\)-primary ideal such that \(I^2={\mathfrak q}I\) for some parameter ideal \({\mathfrak q}=(a_1,a_2,\dots,a_d)A\) contained in \(I\). Suppose that \(I\supseteq \sum(a_1, \dots, \check a_i, \dots, a_d):{\mathfrak m}\). Then \(G(I)_M\) is a Buchsbaum ring and \(H_M^i (G(I))\) is concentrated in degree \(1-i\).
reviewed by: Jun Nishimura
Mathematics Subject Classification ID: 13A30
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 1234252
zbMATH Keywords: \(d^+\)-sequences
zbMATH Keywords: Buchsbaum property
zbMATH Keywords: associated graded ring
MaRDI profile type: Publication
full work available at URL: jabr.1998.7518
OpenAlex ID: W1982623171
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DOI: 10.1006/JABR.1998.7518
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description: scientific article; zbMATH DE number 1234252