Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

Minh, Nguyên Công; Trung, Ngô Viêt

doi:10.1016/j.aim.2010.08.005

Cited by 50 publications

(65 citation statements)

References 11 publications

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“…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.…”

Section: Introductionmentioning

confidence: 98%

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

Minh

2015

Vietnam J. Math.

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Let I be the Stanley-Reisner ideal of a simplicial complex . In this short paper, we shall give a formula of vanishing of the local cohomology modules for S/I (r) in the case is a union of disjoint matroids, where I (r) is the rth symbolic power of I . As an application, we will improve a previous result in Minh and Nakamura (Nagoya Math. J. 213, 127-140, 2014) for the k-Buchsbaumness of S/I (r) .

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“…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.…”

Section: Introductionmentioning

confidence: 98%

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

Minh

2015

Vietnam J. Math.

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“…Recently, Terai and Trung [21] showed that ∆ is a complete intersection whenever I m ∆ is Cohen-Macaulay for some m 3. By contrast with this situation we have not known a characterization of ∆ for which I 2 ∆ is Cohen-Macaulay yet (see [8], [12], [16], [21]). On the other hand, Vasconcelos (see [23,Conjecture (B)]) suggests that ∆ must be Gorenstein if I 2 ∆ is Cohen-Macaulay.…”

Section: Introductionmentioning

confidence: 99%

A characterization of triangle-free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals

Hoang

Trung

2015

J Algebr Comb

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“…(f) ⇒ (b) If Ind(H) is a tight complex, then R/I(H) (2) is Cohen-Macaulay, by [6, Theorem 2.5] (see also [7]), where I(H) (2) is the second symbolic power of I(H). Thus, R/I(H) is also Cohen-Macaulay, by [6, Theorem 2.1].…”

Section: H)mentioning

confidence: 99%

The Edge Ideals of Complete Multipartite Hypergraphs

Kiani

Madani

2015

Communications in Algebra

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Abstract. We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, l-Cohen-Macaulay, l-Buchsbaum, unmixed, and satisfying Serre's condition Sr, via some combinatorial terms. Also, we prove that for a complete s-uniform t-partite hypergraph H, vertex decomposability, shellability, sequentially Sr and sequentially Cohen-Macaulay properties coincide with the condition that H has t−1 sides consisting of a single vertex. Moreover, we show that the latter condition occurs if and only if it is a chordal hypergraph. IntroductionIn recent years, many authors have focused on studying different kinds of monomial ideals associated to combinatorial objects, such as Stanley-Reisner ideals, facet ideals, edge ideals and path ideals (see for example [3], [10], [11], [12], [16] and [17]). In this paper, we study edge ideals of hypergraphs.A hypergraph H with finite vertex set V (H) is a family of nonempty subsets of V (H) whose union is V (H), called edges. The set of vertices and edges of H are denoted by V (H) and E(H), respectively. Sometimes, we also use H as its set of edges. An induced subhypergraph of H, over S ⊆ V (H), is defined as H S = {e ∈ H : e ⊆ S}. If all the edges of a hypergraph H have the same cardinality t, then it is said that H is t-uniform (also sometimes referred as a t-graph). We call an edge of cardinality one, an isolated vertex. If none of the edges of H is included in another, then H is called a simple hypergraph. Throughout this paper, we mean by a hypergraph, a simple one.In this paper, we focus on a class of uniform hypergraphs called t-partite which is a generalization of multipartite graphs. These hypergraphs are important from the combinatorial point of view. The organization of this paper is as follows. In Section 2, we bring some definitions and general properties of hypergraphs and simplicial complexes, and also some relations between combinatorial objects and algebraic ones. For this purpose, we mostly use [1] and [5]. Moreover, we introduce the class of multipartite hypergraphs, and in the case of complete multipartite hypergraphs, we pose some of their basic properties, which will be used in the other sections. In Section 3, we investigate about some algebraic properties of complete multipartite hypergraphs, like being (almost) complete intersection, Gorenstein, level, l-Cohen-Macaulay, l-Buchsbaum, unmixed, and satisfying Serre's condition S r , via studying their independent complexes. For a complete s-uniform t-partite hypergraph H, we show that level, Cohen-Macaulay and S r properties are equivalent to the condition that all sides of H have just one vertex, which happens just in the case that the independent complex of H is a matroid. On the other hand, we show that the independent complex of a complete s-uniform t-partite hypergraph is a matroid if and only if it is a tight complex. Moreover, we show that when s > 2, Buchsbaumness is also equivalent 2010 Mathematics Subject Cla...

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Cohen–Macaulayness of monomial ideals and symbolic powers of Stanley–Reisner ideals

Cited by 50 publications

References 11 publications

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

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

A characterization of triangle-free Gorenstein graphs and Cohen–Macaulayness of second powers of edge ideals

The Edge Ideals of Complete Multipartite Hypergraphs

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