Shedding vertices and Ass-decomposable monomial ideals

Manuscripta Mathematica, 2016

Let ∆ be a simplicial complex. We study the expansions of ∆ mainly to see how the algebraic and combinatorial properties of ∆ and its expansions are related to each other. It is shown that ∆ is Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum or k-decomposable, if and only if an arbitrary expansion of ∆ has the same property. Moreover, some homological invariants like the regularity and the projective dimension of the Stanley-Reisner ideals of ∆ and those of their expansions are compared.

Journal of Pure and Applied Algebra, 2015

The purpose of this note is to study containment relations and asymptotic invariants for ideals of fixed codimension skeletons (simplicial ideals) determined by arrangements of n + 1 general hyperplanes in the n-dimensional projective space over an arbitrary field.

Algebra, Geometry and Their Interactions, 2007

We survey some recent results on the minimal graded free resolution of a square-free monomial ideal. The theme uniting these results is the point-of-view that the generators of a monomial ideal correspond to the maximal faces (the facets) of a simplicial complex ∆. This correspondence gives us a new method, distinct from the Stanley-Reisner correspondence, to associate to a square-free monomial ideal a simplicial complex. In this context, the monomial ideal is called the facet ideal of ∆. Of particular interest is the case that all the facets have dimension one. Here, the simplicial complex is a simple graph G, and the facet ideal is usually called the edge ideal of G. Many people have been interested in understanding how the combinatorial data or structure of ∆ appears in or affects the minimal graded free resolution of the associated facet ideal. In the first part of this paper, we describe the current state-of-the-art with respect to this program by collecting together many of the relevant results. We sketch the main details of many of the proofs and provide pointers to the relevant literature for the remainder. In the second part we introduce some open questions which will hopefully inspire future research on this topic.

2004

Let ∆ be an abstract finite simplicial complex with vertices X1, . . . , Xn. Let k be a field throughout this chapter. Let R denote the polynomial ring k[X1, X2, . . . , Xn], where, by abuse of notation, we regard the vertices X1, X2, . . . , Xn as indeterminates over k. Let I∆ be the ideal of R generated by the monomials Xi1 . . . Xir , i1 < i2 < . . . < ir such that {Xi1 , . . . , Xir} is not a face of ∆. The face ring of ∆ is the quotient ring k[∆] := R/I∆. Since I∆ is a homogeneous ideal, k[∆] is a graded ring. In this section we will prove Stanley’s formula for the Hilbert series of k[∆]. In some sense, this formula opened up the connection of Commutative Algebra with Combinatorics. We will exhibit the power of Hilbert series methods by giving an elementary proof of Dehn-Sommerville equations towards the end of this section. We begin by establishing the primary decomposition of I∆. (1.1) Definition. Let F be a face of a simplicial complex ∆. Let PF denote the prime ide...

arXiv (Cornell University), 2021

Given an arbitrary hypergraph H, we may glue to H a family of hypergraphs to get a new hypergraph H ′ having H as an induced subhypergraph. In this paper, we introduce three gluing techniques for which the topological and combinatorial properties (such as Cohen-Macaulayness, shellability, vertex-decomposability etc.) of the resulting hypergraph H ′ is under control in terms of the glued components. This enables us to construct broad classes of simplicial complexes containing a given simplicial complex as induced subcomplex satisfying nice topological and combinatorial properties. Our results will be accompanied with some interesting open problems. introduction A simplicial complex ∆ on a vertex set V is a collection of subsets of V such that ∪∆ = V and ∆ is closed under the operation of taking subsets. The elements of ∆ are called faces and the maximal faces of ∆, under inclusion, are called the facets of ∆. A simplicial complex with facets F 1 ,. .. , F m is often denoted by F 1 ,. .. , F m. A simplex is a simplicial complex with only one facet. A simplicial complex ∆ is called shellable if there is a total order on facets of ∆, say F 1 ,. .. , F m , such that F 1 ,. .. , F i−1 ∩ F i is generated by a non-empty set of maximal proper subsets of F i for 2 ≤ i ≤ m. The notion of shellability is used to give (an inductive) proof for the Euler-Poincaré formula in any dimension. If f i denotes the number of ifaces of a d-dimensional polytope (with f −1 = f d = 1), then the Euler-Poincaré formula states that d i=−1 (−1) i f i = 1. Shellable complexes are themselves an intermediate family among two other important families of simplicial complexes, namely vertex-decomposable and sequentially Cohen-Macaulay simplicial complexes. Indeed, we have the following implications vertex-decomposable =⇒ shellable =⇒ sequentially Cohen-Macaulay, and both of these inclusions are known to be strict. A vertex-decomposable simplicial complex ∆ is defined recursively in terms of link and deletion of vertices of ∆. In a more general setting, the link and the deletion of a face F of ∆ are defined as follows:

2017

In this paper, we introduce the concept of k-clean monomial ideals as an extension of clean monomial ideals and present some homological and combinatorial properties of them. Using the hierarchal structure of k-clean ideals, we show that a (d-1)-dimensional simplicial complex is k-decomposable if and only if its Stanley-Reisner ideal is k-clean, where k≤ d-1. We prove that the classes of monomial ideals like monomial complete intersection ideals, Cohen-Macaulay monomial ideals of codimension 2 and symbolic powers of Stanley-Reisner ideals of matroid complexes are k-clean for all k≥ 0.

We present an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a complex is grafted. These properties have strong algebraic implications for their corresponding facet ideals. 1 Introduction The main goal of this paper is to demonstrate that it is possible to check, in polynomial time, if a monomial ideal is the facet ideal of a simplicial tree. Facet ideals were introduced in [4] (generalizing [9] and [8]) as a method to study square-free monomial ideals. The idea is to associate a simplicial complex to a square-free monomial ideal, where each facet (maximal face) of the complex is the collection of variables that appear in a monomial in the minimal generating set of the ideal (see Definition 2.4). The ideal will then be called the "facet ideal" of this simplicial complex. Special simplicial complexes are called "simplicial trees" (Definition 2.9). Facet ideals of trees have many properties; for example, they have normal and Cohen-Macaulay Rees rings [4]. Finding such classes of ideals is in general a very difficult problem. Simplicial trees also have very strong Cohen-Macaulay properties: their facet ideals are always sequentially Cohen-Macaulay [6], and one can determine under precisely what combinatorial conditions on the simplicial tree the facet ideal is Cohen-Macaulay [5]. In [7] it is shown that the theory is not restricted to square-free monomial ideals; via polarization, one can extend many properties of facet ideals to all monomial ideals. All these properties, and many others, make simplicial trees extremely useful from an algebraic point of view. But how does one determine if a given square-free monomial ideal is the facet ideal of a simplicial tree? In Section 3, we show that this can be decided in polynomial time. This extended abstract is organized as follows: in Section 2 we introduce the notion of a complex, a tree, and a cycle. Section 3 contains the main theoretical result that enables us to produce a polynomial time algorithm to decide whether a given complex is a tree. The algorithm itself is introduced in Section 3.1, and the complexity and optimizations are discussed in Sections 3.2 and 3.3. Section 4 focuses on the algebraic properties of facet ideals: in Section 4.1 we discuss a method of adding generators to a square-free monomial ideal (or facets to the corresponding complex) so that the resulting facet ideal is Cohen-Macaulay. This method is called "grafting" a simplicial complex. For simplicial trees, being grafted and being Cohen-Macaulay are equivalent conditions [5]. We then introduce an algorithm that checks whether or not a given simplicial complex is grafted in Section 4.2, and discuss its complexity in Section 4.3.

Israel Journal of Mathematics, 1994

We find a decomposition of simplicial complexes that implies and sharpens the characterization (due to Bj6rner and Kalai) of the f-vector and Betti numbers of a simplicial complex. It generalizes a result of Stanley, who proved the a~yclic case, and settles a conjecture of Stanley and Kalai.

Journal of Algebra, 2007

We provide a new combinatorial approach to study the minimal free resolutions of edge ideals, that is, quadratic square-free monomial ideals. With this method we can recover most of the known results on resolutions of edge ideals with fuller generality, and at the same time, obtain new results. Past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or Koszul homology groups. Our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph. We also show that our technique extends successfully to the study of graded Betti numbers of arbitrary square-free monomial ideals viewed as facet ideals of simplicial complexes.

Journal of Algebraic Combinatorics

Let L n be a line graph with n edges and F (L n ) the facet ideal of its matching complex. In this paper, we provide the irreducible decomposition of F (L n ) and some exact formulas for the projective dimension and the regularity of F (L n ).

Advances in Mathematics, 2007

We introduce and study vertex cover algebras of weighted simplicial complexes. These algebras are special classes of symbolic Rees algebras. We show that symbolic Rees algebras of monomial ideals are finitely generated and that such an algebra is normal and Cohen-Macaulay if the monomial ideal is squarefree. For a simple graph, the vertex cover algebra is generated by elements of degree 2, and it is standard graded if and only if the graph is bipartite. We also give a general upper bound for the maximal degree of the generators of vertex cover algebras.

TURKISH JOURNAL OF MATHEMATICS, 2016

To a simplicial complex ∆ , we associate a square-free monomial ideal F (∆) in the polynomial ring generated by its facet over a field. Furthermore, we could consider F(∆) as the Stanley-Reisner ideal of another simplicial complex δN (F(∆)) from facet ideal theory and Stanley-Reisner theory. In this paper, we determine what families of simplicial complexes ∆ have the property that their Stanley-Reisner complexes δN (F(∆)) are shellable. Furthermore, we show that the simplicial complex with the free vertex property is sequentially Cohen-Macaulay. This result gives a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.

Journal of Algebra, 1997

A simplicial poset, a poset with a minimal element and whose every interval is a Boolean algebra, is a generalization of a simplicial complex. Stanley defined a ring A associated with a simplicial poset P that generalizes the face-ring of a P w x simplicial complex. If V is the set of vertices of P, then A is a k V -module; we P find the Betti polynomials of a free resolution of A , and the local cohomology P modules of A , generalizing Hochster's corresponding results for simplicial com-P plexes. The proofs involve splitting certain chain or cochain complexes more finely than in the simplicial complex case. Corollaries are that the depth of A is a P topological invariant, and that the depth may be computed in terms of the Cohen-Macaulayness of skeleta of P, generalizing results of Munkres and Hibi.

2013

In a 2008 paper, the first author and Van Tuyl proved that the regularity of the edge ideal of a graph G is at most one greater than the matching number of G. In this note, we provide a generalization of this result to any square-free monomial ideal. We define a 2-collage in a simple hypergraph to be a collection of edges with the property that for any edge E of the hypergraph, there exists an edge F in the 2-collage such that |E \ F | ≤ 1. The Castelnuovo-Mumford regularity of the edge ideal of a simple hypergraph is bounded above by a multiple of the minimum size of a 2-collage. We also give a recursive formula to compute the regularity of a vertex-decomposable hypergraph. Finally, we show that regularity in the graph case is bounded by a certain statistic based on maximal packings of nondegenerate star subgraphs.