Buchsbaum homogeneous algebras with minimal multiplicity
type: Item
instance of: scholarly article
title: Buchsbaum homogeneous algebras with minimal multiplicity
zbMATH Open document ID: 1121.13025
author: Shiro Goto
author: Ken-Ichi Yoshida
published in: Journal of Pure and Applied Algebra
publication date: 2007-06-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0607415
review text: In the present paper, the authors introduce a new class of graded algebras and study its properties. Let \(A\) be a homogeneous graded algebra over a field and \(\mathfrak m\) the irrelevant ideal of \(A\). The first author [J. Algebra 76, 383--399 (1982; Zbl 0482.13012)] showed that \[ e(A) \geq \sum_{i=1}^{d-1} \ell(H_{\mathfrak m}^i(A)) \tag{1} \] if \(A\) is a Buchsbaum ring where \(e(A)\) is the multiplicity of \(A\), \(d\) the dimension of \(A\) and \(H_{\mathfrak m}^i(M)\) the local cohomology module of \(A\) with respect to \(\mathfrak m\). Furthermore he showed that \(A\) has good properties, for example, \(A\) has a \(2\)-linear free resolution, if the equality of (1) holds. In the paper under review, the authors gave a fine bound of \(e(A)\). Assume that \(A\) is a Buchsbaum ring. Let \(c\) be the codimension of \(A\) and \(q\) the initial degree of \(A\). Then they show that \[ e(A) \geq \binom{c+q-2}{q-2} + \sum_{i=1}^{d-1} \ell(H_{\mathfrak m}^i(A)). \tag{2} \] They also find good properties of \(A\) when the equality of (2) holds.
reviewed by: Takesi Kawasaki
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13H15
zbMATH DE Number: 5167577
zbMATH Keywords: Buchsbaum ring
zbMATH Keywords: linear free resolution
zbMATH Keywords: minimal multiplicity
MaRDI profile type: Publication
OpenAlex ID: W1995356293
arXiv ID: math/0607415
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DOI: 10.1016/J.JPAA.2006.11.004
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description: scientific article