The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1
type: Item
instance of: scholarly article
title: The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1
zbMATH Open document ID: 1218.13013
author: Yukio Nakamura
author: Nguyên Công Minh
published in: Journal of Pure and Applied Algebra
publication date: 2010-10-28T00:00:00Z
review text: Let \(\Delta\) be a pure \(1\)-dimensional simplicial complex on a vertex set \([n]=\{1,2,\dots,n\}\). Let \(S=k[x_1,\dots,x_n]\) be a standard graded polynomial ring over an infinite field. We consider the Stanley-Reisner ring \(S/I\), where \(I=I_\Delta\) is the Stanley-Reisner ideal. It is known that \(S/I\) is always Buchsbaum and it is Cohen-Macaulay if and only if \(\Delta\) is connected. In this paper, the authors consider the problem to characterize Buchsbaumness of \(S/I^{(r)}\) for \(r\in\mathbb N\), where \(I^{(r)}\) denotes the \(r\)th symbolic power of \(I\). The main result (Theorem~3.7) states that (1) \(S/I^{(2)}\) is Buchsbaum if and only if \(\Delta\) is connected, (2) \(S/I^{(3)}\) is Buchsbaum if and only if \(\text{diam}(\Delta)\leq 2\), and (3) for \(r>3\) if \(S/I^{(r)}\) is Buchsbaum, then it is Cohen-Macaulay. Here we mean by \(\text{diam}(\Delta)\) the diameter of \(\Delta\), which means the maximal value of the distance between any two vertices in \(\Delta\). On the other hand, \(S/I^{(r)}\) is of dimension~2 and generalized Cohen-Macaulay, i.e., the local cohomology module \(H^1_{\mathfrak m}(S/I^{(r)})\) is of finite length, where \(\mathfrak m=(x_1,\dots,x_m)S\) is the homogeneous maximal ideal. Thus there exists \(k\in\mathbb N\) such that \(\mathfrak m^kH^1_{\mathfrak m}(S/I^{(r)})=0\). The authors computes the smallest such number \(k\) (Theorem~3.8).
reviewed by: Yukihide Takayama
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 05E45
Mathematics Subject Classification ID: 13D45
Mathematics Subject Classification ID: 05C90
zbMATH DE Number: 5807457
zbMATH Keywords: Stanley-Reisner rings
zbMATH Keywords: Buchsbaum rings
zbMATH Keywords: symbolic powers
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.04.007
OpenAlex ID: W1998074915
cites work: Q4274969
cites work: On local cohomology of a tetrahedral curve
cites work: Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals
cites work: Q3744270
cites work: Q5475907
DOI: 10.1016/J.JPAA.2010.04.007
description: scientific article; zbMATH DE number 5807457