Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

In this paper we extend one direction of Fröberg?s theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and we also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.     

Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d?=?2), and strictly contains the class of ‘strongly stable’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.     

Resolutions of letterplace ideals of posets

We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than c chains, we show that the Betti numbers may be computed from simplicial complexes of no more than c vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of P has tree structure.     

Constructible Ideals

Based on the study of simplicial complexes, one may naturally define the constructible monomial ideals. We connect the square-free constructible ideal with the Stanley–Reisner ideal of the Alexander dual associated to a constructible simplicial complex. We give some properties of constructible ideals, and we compute the Betti numbers. We prove that all monomial ideals with linear quotients are constructible ideals. We also show that all constructible ideals have a linear resolution.     

Cellular resolutions of ideals defined by nondegenerate simplicial homomorphisms

In this paper we introduce the class of ordered homomorphism ideals and prove that these ideals admit minimal cellular resolutions constructed as homomorphism complexes. Motivated by work of Dochtermann and Engström, we introduce the class of cointerval simplicial complexes and investigate their combinatorial and topological properties. As a concrete illustration of these structural results, we introduce and study nonnesting monomial ideals, an interesting family of combinatorially defined ideals.     

Buchsbaum Stanley–Reisner Rings and Cohen–Macaulay Covers

First, we give a new criterion for Buchsbaum Stanley–Reisner rings to have linear resolutions. Next, we prove that every (d ? 1)-dimensional complex Δ of initial degree d is contained in the same dimensional Cohen–Macaulay complex whose (d ? 1)th reduced homology is isomorphic to that of Δ. We call such a simplicial complex a Cohen–Macaulay cover of Δ. And we also show that all the intermediate complexes between Δ and its Cohen–Macaulay cover are Buchsbaum provided that Δ is Buchsbaum. As an application, we determine the h-vectors of the 3-dimensional Buchsbaum Stanley–Reisner rings with initial degree 3.     

Finite Free Resolutions and 1-Skeletons of Simplicial Complexes

A technique of minimal free resolutions of Stanley—Reisner rings enables us to show the following two results: (1) The 1-skeleton of a simplicial (d–1)-sphere is d-connected, which was first proved by Barnette; (2) The comparability graph of a non-planar distributive lattice of rank d–1 is d-connected.     

Minimal Resolutions and the Homology of Matching and Chessboard Complexes

We generalize work of Lascoux and Józefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras. Our motivation is two-fold: We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Björner, Lovasz, Vreica and ivaljevi. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes. The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers.     

Cohen-Macaulay, shellable and unmixed clutters with a perfect matching of König type

Let C be a clutter with a perfect matching e₁,…,e_g of König type and let Δ_C be the Stanley-Reisner complex of the edge ideal of C. If all c-minors of C have a free vertex and C is unmixed, we show that Δ_C is pure shellable. We are able to describe, in combinatorial and algebraic terms, when Δ_C is pure. If C has no cycles of length 3 or 4, then it is shown that Δ_C is pure if and only if Δ_C is pure shellable (in this case e_i has a free vertex for all i), and that Δ_C is pure if and only if for any two edges f₁,f₂ of C and for any e_i, one has that f₁∩e_i⊂f₂∩e_i or f₂∩e_i⊂f₁∩e_i. It is also shown that this ordering condition implies that Δ_C is pure shellable, without any assumption on the cycles of C. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen-Macaulay, and they have linear resolutions. Furthermore if C is admissible and complete, then C is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi-on the structure of unmixed simplicial trees-to clutters with the König property without 3-cycles or 4-cycles.     

Generic initial ideals and squeezed spheres

In 1988 Kalai constructed a large class of simplicial spheres, called squeezed spheres, and in 1991 presented a conjecture about generic initial ideals of Stanley-Reisner ideals of squeezed spheres. In the present paper this conjecture will be proved. In order to prove Kalai's conjecture, based on the fact that every squeezed (d−1)-sphere is the boundary of a certain d-ball, called a squeezed d-ball, generic initial ideals of Stanley-Reisner ideals of squeezed balls will be determined. In addition, generic initial ideals of exterior face ideals of squeezed balls are determined. On the other hand, we study the squeezing operation, which assigns to each Gorenstein* complex Γ having the weak Lefschetz property a squeezed sphere Sq(Γ), and show that this operation increases graded Betti numbers.     

Regularity 3 in edge ideals associated to bipartite graphs

We focus in this paper on edge ideals associated to bipartite graphs and give a combinatorial characterization of those having regularity 3. When the regularity is strictly bigger than 3, we determine the first step i in the minimal graded free resolution where there exists a minimal generator of degree >i+3, show that at this step the highest degree of a minimal generator is i+4, and determine the corresponding graded Betti number β _i,i+4 in terms of the combinatorics of the graph. The results are then extended to the non-square-free case through polarization. We also study a family of ideals of regularity 4 that play an important role in our main result and whose graded Betti numbers can be completely described through closed combinatorial formulas.     

Edge domatic numbers of complete n- partite graphs

An edge dominating set of a graph is a set of edgesD such that every edge not inD is adjacent to an edge inD. An edge domatic partition of a graph G =(V, E) is a collection of pairwise disjoint edge dominating sets of G whose union isE. The maximum size of an edge domatic partition of G is called the edge domatic number of G. In this paper we study the edge domatic numbers of completen-partite graphs. In particular, we give exact values for the edge domatic numbers of complete 3-partite graphs and balanced complete n-partite graphs with oddn.     

Balanced complexes and complexes without large missing faces

The face numbers of simplicial complexes without missing faces of dimension larger than i are studied. It is shown that among all such (d−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal f-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal h-vector. It is also verified that the l-skeleton of a flag (d−1)-dimensional 2-CM complex is 2(d−l)-CM, while the l-skeleton of a flag piecewise linear (d−1)-sphere is 2(d−l)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.     

Decompositions of two-dimensional simplicial complexes

We show that the class of Cohen-Macaulay complexes, that of complexes with constructible subdivisions, and that of complexes with shellable subdivisions differ from each other in every dimension d?2. Further, we give a characterization of two-dimensional simplicial complexes with shellable subdivisions, and show also that they are constructible.     

Taylor complexes for Koszul boundaries

For all boundary modules of the Koszul complex of a monomial sequence we construct complexes, which we call Taylor complexes. For a monomial d-sequences these complexes provide free resolutions of the boundary modules. Let M be the ideal generated by a monomial d-sequence. We use the Taylor complexes to construct minimal free resolutions of the Rees algebra and the associated graded ring of M. Received: 13 November 1997 / Revised version: 6 March 1998     

Regularity of path ideals of gap free graphs

In this paper we study the Castelnuovo–Mumford regularity of the path ideals of finite simple graphs. We find new upper bounds for various path ideals of gap free graphs. In particular we prove that the t-path ideals of gap free, claw free and whiskered-$K_4$ free graphs have linear minimal free resolutions for all $t\ge 3$.     

Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.     

On a conjecture by Kalai

We show that monomial ideals generated in degree two satisfy a conjecture by Eisenbud, Green and Harris. In particular, we give a partial answer to a conjecture of Kalai by proving that h-vectors of flag Cohen-Macaulay simplicial complexes are h-vectors of Cohen-Macaulay balanced simplicial complexes.