A note on the \(k\)-Buchsbaum property of symbolic powers of Stanley-Reisner ideals
type: Item
instance of: scholarly article
title: A note on the \(k\)-Buchsbaum property of symbolic powers of Stanley-Reisner ideals
zbMATH Open document ID: 1235.13017
author: Yukio Nakamura
author: Nguyên Công Minh
published in: Tokyo Journal of Mathematics
publication date: 2011-09-27T00:00:00Z
review text: Let \(S=F[x_1,\dots,x_n]\) be a polynomial ring over a field \(F.\) Let \(S/I_{\Delta}\) be the Stanley-Reisner ring associated to a pure simplicial complex \(\Delta,\) of dimension \(1,\) on the vertex set \(\{1,\dots,n\}.\) The goal of the paper is to study the \(k\)-Buchsbaum property of \(S/I^{(r)},\) where \(I^{(r)}\) is the \(r\)-th symbolic power of \(I=I_{\Delta}.\) The condition for \(S/I^{(r)}\) to be \(k\)-Buchsbaum is equivalent to saying that \(k\) is the minimal number satisfying \(m^kH^1_m(S/I^{(r)})=0,\) where \(m=(x_1,\dots,x_n)+I.\) In the main theorem the function \[ k(r):=\min\{k\mid m^kH^1_m(S/I^{(r)}=0\} \] is computed when \(\Delta\) is not connected. The connected case was solved in a previous paper of the same authors.
Mathematics Subject Classification ID: 13F55
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 05E99
zbMATH DE Number: 5950742
zbMATH Keywords: Stanley-Reisner
zbMATH Keywords: \(k\)-Buchsbaum
zbMATH Keywords: monomial ideal
zbMATH Keywords: connected graph
zbMATH Keywords: symbolic power
zbMATH Keywords: Cohen-Macaulay
reviewed by: Giuseppe Zappalà
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1313074452
OpenAlex ID: W2043018683
cites work: Q3843140
cites work: The Buchsbaum property of symbolic powers of Stanley-Reisner ideals of dimension 1
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: Q5475907
DOI: 10.3836/TJM/1313074452
description: scientific article; zbMATH DE number 5950742