Bounds for the minimum numbers of generators of generalized Cohen- Macaulay ideals
type: Item
instance of: scholarly article
title: Bounds for the minimum numbers of generators of generalized Cohen- Macaulay ideals
zbMATH Open document ID: 0547.13015
DOI: 10.1016/0021-8693(84)90193-5
author: Trung Ngo Viet
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: An ideal I of a local ring (R,\({\mathfrak m})\) is called a generalized Cohen- Macaulay ideal if \(S=R/I\) is a generalized Cohen-Macaulay ring, i.e. \(\ell (H^ i_{{\mathfrak m}}(S))<\infty\) for \(i=0,...,\dim S-1.\) The class of such ideals is rather large, e.g. it includes the defining prime ideals of isolated singularities. Inspired of an article of \textit{J. Sally} on Cohen-Macaulay ideals [Pac. J. Math. 63, 517-520 (1976; Zbl 0314.13009)], this paper will give bounds for the minimum number of generators v(I) of a generalized Cohen-Macaulay ideal I in terms of the embedding dimensions, the dimensions, the multiplicities of R, S, and \({\mathcal L}(H^ i_{{\mathfrak m}}(S))\). There are some interesting applications. For example, if I is the defining prime ideal of an arithmetically Buchsbaum curve C in \(P^ 3_ k\), i.e. \(H^ 1_ m(S)\) is a vector space over the field k, where S is the local ring of the vertex of the affine cone over C, then \(v(I)\leq 3e/2+1,\) where e denotes the degree of C.
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 14M05
zbMATH DE Number: 3871524
zbMATH Keywords: multiplicity
zbMATH Keywords: local cohomology modules
zbMATH Keywords: generalized Cohen-Macaulay ideal
zbMATH Keywords: minimum number of generators
zbMATH Keywords: embedding dimensions
zbMATH Keywords: arithmetically Buchsbaum curve
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90193-5
OpenAlex ID: W2020603903
cites work: Local Rings of High Embedding Dimension
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cites work: Q3913515
cites work: Verallgemeinerte COHEN-MACAULAY-Moduln
cites work: Prestable ideals
cites work: Bounds for numbers of generators of Cohen-Macaulay ideals
cites work: Q3849050
description: scientific article; zbMATH DE number 3871524