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

Abstract

Let S = k[x 1 , x 2 , . . . , x n ] be a polynomial ring. Let I be a StanleyReisner ideal in S of a pure simplicial complex of dimension one. In this paper, we study the Buchsbaum property of S/I r for any integer r > 0. Our first purpose is giving a characterization of Ext-modules Ext p S (S/m t , S/ J ) for any monomial ideal J , where m t = (x t 1 , x t 2 , . . . , x t n ), in terms of certain simplicial complexes. Then we consider the Buchsbaum property of S/I r . The main tool to check the Buchsbaumness is the surjectivity criterion. We see the behavior of the canonical map from Ext p S (S/m t , S/I r ) to H p m (S/I r ) from the view point of reduced cohomology groups of simplicial complexes.

On the k-Buchsbaum property of powers of Stanley–Reisner ideals

“…Proof. The equivalence of (1), (2), and (4) follows from [MT1,Corollary 3.4] and [MN2,Theorem 4.10]. The implication (1) ⇒ (3) follows from Proposition 3.4 and Lemma 4.1.…”

“…Research on the Cohen-Macaulay property of S/I (t) Δ and S/I t Δ was begun by [MT1] and [GH] for 1-dimensional simplicial complexes Δ. Then, by the authors, the Buchsbaum properties of S/I (t) Δ and S/I t Δ were studied in [MN1] and [MN2] for Δ with dim Δ = 1. Moreover, the k-Buchsbaum property of S/I (t) Δ was studied in [MN3].…”

“…In Section 2, we set up the fundamental notation and terminologies, for which we mainly refer to the book [BH]. Degree complex Δ a (I) plays an important role (for details, we refer to [T], [MN1], and [MN2]). In Section 3, we provide an argument for the symbolic powers of Stanley-Reisner ideals.…”

Geometric regularity of powers of two-dimensional squarefree monomial ideals

“…Two-dimensional squarefree monomial ideals attract many authors' interests. For example, the Buchsbaum property of symbolic powers and ordinary powers of these ideals was studied in [15] and [16], respectively, and the Cohen-Macaulayness of symbolic powers and ordinary powers of such ideals was characterized in terms of the properties of their associated graphs in [17]. Recently, the regularity of symbolic powers of such ideals was computed explicitly in [12].…”

Combinatorial characterizations of the Cohen-Macaulayness of the second power of edge ideals

“…The above questions are of interest for what have been mentioned at the beginning, and here we answer them under the additional assumption that ∆ is a flag simplicial complex. Let us consider the first one: An answer to this question is already given in [MT2,Theorem 2.1] without the flag condition of ∆ (also see [MN1] and [MN2] for some special cases of the second part of this question). However, [MT2,Theorem 2.1] does not give a characterization in a combinatorial fashion, rather it involves the Cohen-Macaulayness of a family F of certain subcomplexes of ∆.…”