Buchsbaumness of ordinary powers of two-dimensional square-free monomial ideals
type: Item
instance of: scholarly article
title: Buchsbaumness of ordinary powers of two-dimensional square-free monomial ideals
zbMATH Open document ID: 1220.13015
author: Nguyên Công Minh
author: Yukio Nakamura
published in: Journal of Algebra
publication date: 2011-05-16T00:00:00Z
review text: Let \(S=k[x_1,\dots,x_n]\) denote the polynomial ring in \(n\) variables of a field \(k\). Let \(I\) denote the Stanley-Reisner ring of a pure one-dimensional simplicial complex \(\Delta\). The subject of this paper is the investigation when \(S/I^r\) is a Buchsbaum ring. The following results are shown: (1) Let \(n\geq 4\) and \(r\geq 4\), then \(S/I^r\) is a Buchsbaum ring if and only if \(\Delta\) is a square. (2) Let \(n\geq 4\), then \(S/I^3\) is a Buchsbaum ring is and only if \(\Delta\) is a square or a pentagon. (3) With the notation above and \(n\geq 4\) the following conditions are equivalent: (i) \(S/I^2\) is Buchsbaum. (ii) \(\Delta\) is connected, does not contain any triangle of edges, and is claw-free (that is \(\Delta\) has no induced bipartite subgraph with one vertex on one side and three vertices on the other). (iii) \(\Delta\) is a path of length \(n-1\) or a cycle of length \(n\). The main technical tool for the proof is a description of \(\text{Ext}_S^p(S/\mathfrak m_t,S/J)\), where \(\mathfrak m_t=(x_1^t,\dots,x_n^t)S\) and \(J\) is a monomial ideal, in terms of certain simplicial complexes. This is used for proving the Buchsbaum property by the subjectivity criterion. The statements in (1) and (2) are generalizations of corresponding results for Cohen-Macaulay rings shown by \textit{N. C. Minh} and \textit{N. V. Trung} [J. Algebra 322, No. 12, 4219--4227 (2009; Zbl 1206.13028)].
reviewed by: Peter Schenzel
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 05E45
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13A15
zbMATH DE Number: 5888180
zbMATH Keywords: Buchsbaum ideal
zbMATH Keywords: monomial ideal
zbMATH Keywords: symbolic power
zbMATH Keywords: claw-free
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2010.07.027
OpenAlex ID: W1973082700
cites work: Q3843140
cites work: Characterizations of Buchsbaum complexes
cites work: Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals
cites work: On the number of faces of simplicial complexes and the purity of Frobenius
cites work: Q3744270
cites work: Q5475907
cites work: Locally complete intersection Stanley-Reisner ideals
cites work: Q4528984
DOI: 10.1016/J.JALGEBRA.2010.07.027
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description: scientific article; zbMATH DE number 5888180