The \(k\)-Buchsbaum property for some polynomial ideals
type: Item
instance of: scholarly article
title: The \(k\)-Buchsbaum property for some polynomial ideals
zbMATH Open document ID: 1076.13013
author: H. Bresinsky
author: Le Tuan Hoa
published in: Journal of Mathematics of Kyoto University
publication date: 2005-01-03T00:00:00Z
review text: Let \(R\) be a standard graded ring over a field \(K\), and let \(\mathfrak m\) be the maximal homogeneous ideal. A graded module \(M\) over \(R\) is said to be \(k\)-Buchsbaum if its local cohomology modules \(H^i_{\mathfrak m} (M)\), \(0 \leq i \leq d\), are annihilated by \({\mathfrak m}^k\), where \(d+1\) is the Krull dimension of \(M\). In this article, the authors focus on the case where \(R = R_{r+1} := K[x_0,\dots,x_r]\) and \(M = R/\mathfrak a\), where \(\mathfrak a\) is a homogeneous ideal and \(K\) is infinite. An equivalent (and, according to the authors, more workable) definition for \(k\)-Buchsbaum ideals, and some technical reformulations, is given in terms of systems of parameters. Much earlier, \textit{H. Bresinsky} and \textit{W. Vogel} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 39, 143--159 (1993; Zbl 0837.13008)] had given an algorithm to test if such an ideal is Cohen-Macaulay or Buchsbaum. (The former corresponds to \(k=0\); the latter means that \(R_{r+1}/{\mathfrak a}\) and \(R_{r+1}/({\mathfrak a},F_0,\dots,F_i )\), \(0 \leq i \leq d\), are 1-Buchsbaum for any system of parameters.) The central problem addressed in this paper is to find an algorithm to determine, or at least bound from above, the \(k\)-Buchsbaumness of an ideal, for larger \(k\), without explicit computation of Ext-modules or local cohomology modules. The authors are successful in the case of certain binomial ideals, using Gröbner bases computations, and in fact they find the smallest such \(k\).
reviewed by: Juan C. Migliore
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 13D45
zbMATH DE Number: 2124474
zbMATH Keywords: \(k\)-Buchsbaum module
zbMATH Keywords: Cohen-Macaulay property
zbMATH Keywords: local cohomology
zbMATH Keywords: Gröbner basis
zbMATH Keywords: system of parameters
zbMATH Keywords: binomial ideals
zbMATH Keywords: \(k\)-Buchsbaum ideals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250281731
OpenAlex ID: W1574489400
DOI: 10.1215/KJM/1250281731
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description: scientific article; zbMATH DE number 2124474