The Buchsbaum property of symbolic powers of Stanley–Reisner ideals of dimension 1

Abstract

a b s t r a c tLet S be a polynomial ring and I be the Stanley-Reisner ideal of a simplicial complex ∆. The purpose of this paper is investigating the Buchsbaum property of S/I (r) when ∆ is pure dimension 1. We shall characterize the Buchsbaumness of S/I (r) in terms of the graphical property of ∆. That is closely related to the characterization of the Cohen-Macaulayness of S/I (r) due to the first author and N.V. Trung. Crown

Buchsbaumness of ordinary powers of two-dimensional square-free monomial ideals

“…The classification is given as follows. We note that the condition of Theorem 4.10 is equivalent to saying that S/I r is Cohen-Macaulay for some r 3 by [MT,Corollary 3.5]. …”

“…We note that the symbolic power I (t) of I is equal to I (t) = {i, j}∈ P t ij . The Cohen-Macaulay property of S/I t and S/I (t) are studied in [MT,Theorems 2.3 and 2.4]. The Buchsbaum property for S/I (t) is studied in [MN,Theorem 3.7].…”

“…The Cohen-Macaulay property of S/I t and S/I (t) are studied in [MT,Theorems 2.3 and 2.4]. The Buchsbaum property for S/I (t) is studied in [MN,Theorem 3.7]. In order to study the Buchsbaum property of S/I t , we pick up important results stated in [MT,MN], which will be applied several times in our argument.…”

“…We first note that is connected since S/I (r) is Buchsbaum by Lemma 4.4 and [MN,Theorem 3.7]. Suppose that K V ⊆ for some V ⊆ [n] with |V | = 3.…”

A Remark on the Local Cohomology Modules of a Union of Disjoint Matroids

“…In Section 2, we set up notations, terminologies. We quote some fundamental results from [1,8] and [4]. In Section 3, we shall give the proof of the main result.…”