Free Resolutions of Simplicial Posets

Communications in Algebra, 2016

We use the natural homeomorphism between a regular CW-complex X and its face poset P X to establish a canonical isomorphism between the cellular chain complex of X and the result of applying the poset construction of [Cla10] to P X. For a monomial ideal whose free resolution is supported on a regular CW-complex, this isomorphism allows the free resolution of the ideal to be realized as a CW-poset resolution. Conversely, any CW-poset resolution of a monomial ideal gives rise to a resolution supported on a regular CW-complex.

Bulletin of the Iranian Mathematical Society, 2018

For a simplicial complex ∆, the affect of the expansion functor on combinatorial properties of ∆ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal I(∆) and its Alexander dual which we denote by J∆ to see how the expansion functor alter the algebraic properties of these ideals. It is shown that for any expansion ∆ α the ideals J∆ and J∆α have the same total Betti numbers and their Cohen-Macaulayness are equivalent, which implies that the regularities of the ideals I(∆) and I(∆ α) are equal. Moreover, the projective dimensions of I(∆) and I(∆ α) are compared. In the sequel for a graph G, some properties that are equivalent in G and its expansions are presented and for a Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable) graph G, we give some conditions for adding or removing a vertex from G, so that the remaining graph is still Cohen-Macaulay (resp. sequentially Cohen-Macaulay and shellable).

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...

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.

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.

Arkiv för Matematik, 2012

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CMt, is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen-Macaulay and k-Buchsbaum. In analogy with the Cohen-Macaulay and Buchsbaum complexes, we give some characterizations of CMt (=1−CMt) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CMt property under the operation of join of two simplicial complexes. We show that a complex is k-CMt if and only if the links of its non-empty faces are k-CM t−1. We prove that for an integer s≤d, the (d−s−1)-skeleton of a (d−1)-dimensional k-CMt complex is (k+s)-CMt. This result generalizes Hibi's result for Cohen-Macaulay complexes and Miyazaki's result for Buchsbaum complexes.

2013

In this paper we study the Alexander dual of a vertex decomposable simplicial complex. We define the concept of a vertex splittable ideal and show that a simplicial complex $\Delta$ is vertex decomposable if and only if $I_{\Delta^{\vee}}$ is a vertex splittable ideal. Moreover, the properties of vertex splittable ideals are studied. As the main result, it is proved that any vertex splittable ideal has a Betti splitting and the graded Betti numbers of such ideals are explained with a recursive formula. As a corollary, recursive formulas for the regularity and projective dimension of $R/I_{\Delta}$, when $\Delta$ is a vertex decomposable simplicial complex, are given. Moreover, for a vertex decomposable graph $G$, a recursive formula for the graded Betti numbers of its vertex cover ideal is presented. In special cases, this formula is explained, when $G$ is chordal or a sequentially Cohen-Macaulay bipartite graph. Finally, among the other things, it is shown that an edge ideal of a graph is vertex splittable if and only if it has linear resolution.

Journal of Pure and Applied Algebra, 1998

Associated to any simplicial complex A on n vertices is a square-free monomial ideal I.1 in the polynomial ring A = k[x~, ,x,1, and its quotient k[A] = A/IA known as the Stanley-Reisner ring. This note considers a simplicial complex A* which is in a sense a canonical Alexander dual to A, previously considered in [I, 51. Using Alexander duality and a result of Hochster computing the Betti numbers dimkTor,A(k[d],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in A*. As corollaries, we prove that 1~ has a linear resolution as A-module if and only if A* is Cohen-Macaulay over k. and show how to compute the Betti numbers dimkTort(k[A],k)

arXiv (Cornell University), 2020

This paper concerns the study of a class of clutters called simplicial subclutters. Given a clutter C and its simplicial subclutter D, we compare some algebraic properties and invariants of the ideals I, J associated to these two clutters, respectively. We give a formula for computing the (multi)graded Betti numbers of J in terms of those of I and some combinatorial data about D. As a result, we see that if C admits a simplicial subclutter, then there exists a monomial u / ∈ I such that the (multi)graded Betti numbers of I + (u) can be computed through those of I. It is proved that the Betti sequence of any graded ideal with linear resolution is the Betti sequence of an ideal associated to a simplicial subclutter of the complete clutter. These ideals turn out to have linear quotients. However, they do not form all the equigenerated square-free monomial ideals with linear quotients. If C admits ∅ as a simplicial subclutter, then I has linear resolution over all fields. Examples show that the converse is not true.

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.

With any locally finite partially ordered set K its incidence algebra Ω(K) is associated. We shall consider algebras over fields with characteristic zero. In this case there is a correspondence K ↔ Ω(K) such that the poset K can be reconstructed from its incidence algebra up to an isomorphism-due to Stanley theorem. In the meantime, a monotone mapping between two posets in general induces no homomorphism of their incidence algebras. In this paper I show that if the class of posets is confined to simplicial complexes then their incidence algebras acquire the structure of differential moduli and the correspondence K ↔ Ω is a contravariant functor.

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.

Discrete & Computational Geometry, 1996

The notion of a partitionable simplicial complex is extended to that of a signable partially ordered set. It is shown in a unified way that face lattices of shellable polytopal complexes, polyhedral cone fans, and oriented matroid polytopes, are all signable. Each of these classes, which are believed to be mutually incomparable, strictly contains the class of convex polytopes. A general sufficient condition, termed total signability, for a simplicial complex to satisfy McMullen's Upper Bound Theorem on the numbers of faces, is provided. The simplicial members of each of the three classes above are concluded to be partitionable and to satisfy the upper bound theorem. The computational complexity of face enumeration and of deciding partitionability is discussed. It is shown that under a suitable presentation, the face numbers of a signable simplicial complex can be efficiently computed. In particular, the face numbers of simplicial fans can be computed in polynomial time, extending the analogous statement for convex polytopes.

Progress in Commutative Algebra 1, 2012

We give a brief survey of the various topological and combinatorial techniques which have been used to construct the minimal free resolution of a stable monomial ideal in a polynomial ring over a field. The new results appearing in this paper describe a connection between certain topological and combinatorial methods for the description of said minimal resolutions. In particular, we construct a minimal poset resolution of an arbitrary stable monomial ideal by using a poset of Eliahou-Kervaire admissible symbols associated to a stable ideal. The structure of the poset under consideration is quite rich and in related analysis, we exhibit a regular CW complex which supports this resolution.

The Bulletin of the Malaysian Mathematical Society Series 2

For a positive integer k a class of simplicial complexes, to be denoted by CM(k), is introduced. This class generalizes Cohen-Macaulay simplicial complexes. In analogy with the Cohen-Macaulay complexes, we give some homological and combinatorial properties of CM(k) complexes. It is shown that the complex ∆ is CM(k) if and only if I ∆ ∨ , the Stanley-Reisner ideal of the Alexander dual of ∆, has a k-resolution, i.e. β i. j (I ∆ ∨) = 0 unless j = ik + q, where q is the degree of I ∆ ∨. As a main result, we characterize all bipartite graphs whose independence complexes are CM(k) and show that an unmixed bipartite graph is CM(k) if and only if it is pure k-shellable. Our result improves a result due to Herzog and Hibi and also a result due to Villarreal.