Perfect ideals of grade three defined by skew-symmetrizable matrices
type: Item
instance of: scholarly article
zbMATH Open document ID: 1248.13015
DOI: 10.4134/BKMS.2012.49.4.715
author: Yong Sung Cho
author: Oh-Jin Kang
author: Hyoung June Ko
published in: Bulletin of the Korean Mathematical Society
publication date: 2012-08-16T00:00:00Z
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 13D40
Mathematics Subject Classification ID: 13C40
zbMATH DE Number: 6066740
zbMATH Keywords: skew-symmetrizable matrix
zbMATH Keywords: generalized alternating matrix
zbMATH Keywords: almost complete intersection of grade three
zbMATH Keywords: perfect ideal of grade three
zbMATH Keywords: minimal free resolution
zbMATH Keywords: linkage
zbMATH Keywords: Hilbert function
MaRDI profile type: Publication
OpenAlex ID: W2092358528
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title: Perfect ideals of grade three defined by skew-symmetrizable matrices
review text: A square matrix \(M\) is said to be skew-symmetrizable or generalized alternating if there exist nonzero diagonal matrices \(C,D\) such that \(CMD\) is an alternating matrix. The notion of a skew-symmetrizable matrix was introduced by authors in order to describe a structure theorem for complete intersections of grade 4 [\textit{O.-J. Kang} and \textit{H. Ko}, Algebra Colloq.\ 12, No. 2, 181--197 (2005; Zbl 1100.13012)].NEWLINENEWLINEThe purposes of the paper are to construct a skew-symmetrizable matrix \(G_3\) of degree \(r+4\) (Definition 3.6) and an ideal \(I=\overline{Pf_{r+3}(G_3)}\) associated with \(G_3\) (Definition 3.12), to show Theorem 3.17 (see below), and to compute the Hilbert function of \(R/I\) in the case where \(I\) is a homogeneous perfect ideal of grade 3 in a polynomial ring \(R\) over a field (Proposition 4.2). Theorem 3.17: Let \((R,\mathfrak m)\) be a Noetherian local ring and let \(r\) be an odd integer \(>1\). Assume that the entries of matrices defining \(G_3\) are all contained in \(\mathfrak m\). If \(I\) has grade 3, then it is a perfect ideal of type 2.
reviewed by: Yoichi Aoyama
description: scientific article; zbMATH DE number 6066740