On the computation of a-invariants

Thea-invariant of a graded Cohen-Macaulay ring is the least degree of a generator of its graded canonical module. We compute thea-invariants of (i) graded algebras with straightening laws on upper semi-modular lattices and (ii) the Stanley-Reisner rings of shellable weighted simplicial complexes. The formulas obtained are applied to rings defined by determinantal and pfaffian ideals.     

Shellable graphs and sequentially Cohen-Macaulay bipartite graphs

Associated to a simple undirected graph G is a simplicial complex ΔG whose faces correspond to the independent sets of G. We call a graph G shellable if ΔG is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we classify all the shellable bipartite graphs; they are precisely the sequentially Cohen-Macaulay bipartite graphs. We also give a recursive procedure to verify if a bipartite graph is shellable. Because shellable implies that the associated Stanley-Reisner ring is sequentially Cohen-Macaulay, our results complement and extend recent work on the problem of determining when the edge ideal of a graph is (sequentially) Cohen-Macaulay. We also give a new proof for a result of Faridi on the sequentially Cohen-Macaulayness of simplicial forests.     

The facet ideal of a simplicial complex

 To a simplicial complex, we associate a square-free monomial ideal in the polynomial ring generated by its vertex set over a field. We study algebraic properties of this ideal via combinatorial properties of the simplicial complex. By generalizing the notion of a tree from graphs to simplicial complexes, we show that ideals associated to trees satisfy sliding depth condition, and therefore have normal and Cohen-Macaulay Rees rings. We also discuss connections with the theory of Stanley-Reisner rings. Received: 7 January 2002 / Revised version: 6 May 2002     

Glicci simplicial complexes

This note is a case study for the potential of liaison-theoretic methods to applications in Combinatorics. One of the main open questions in liaison theory is whether every homogeneous Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the G-liaison class of a complete intersection. We give an affirmative answer to this question for Stanley-Reisner ideals defined by simplicial complexes that are weakly vertex-decomposable. This class of complexes includes matroid, shifted and Gorenstein complexes respectively. Moreover, we construct a simplicial complex which shows that the property of being glicci depends on the characteristic of the base field. As an application of our methods we establish new evidence for two conjectures of Stanley on partitionable complexes and Stanley decompositions.     

A combinatorial decomposition of simplicial complexes

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

Initial complex associated to a jet scheme of a determinantal variety

We show in this paper that the principal component of the first-order jet scheme over the classical determinantal variety of m×n matrices of rank at most 1 is arithmetically Cohen-Macaulay, by showing that an associated Stanley-Reisner simplicial complex is shellable.     

An alternative proof of a characterization of Cohen-Macaulay bipartite graphs

Recently, Herzog and Hibi explicitly described all Cohen-Macaulay bipartite graphs by using the Stanley-Reisner ideal of the Alexander dual of the simplicial complex Δ _P associated to a finite poset P. In this paper, we will present a short proof that does not use the Stanley-Reisner ideal of the Alexander dual of Δ _P .     

Deficiently extremal Cohen-Macaulay algebras

The aim of this paper is to study homological properties of deficiently extremal Cohen-Macaulay algebras. Eagon-Reiner showed that the Stanley-Reisner ring of a simplicial complex has a linear resolution if and only if the Alexander dual of the simplicial complex is Cohen-Macaulay. An extension of a special case of Eagon-Reiner theorem is obtained for deficiently extremal Cohen-Macaulay Stanley-Reisner rings.     

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.     

Cubical Subdivisions and Local h-Vectors

Face numbers of triangulations of simplicial complexes were studied by Stanley by use of his concept of a local h-vector. It is shown that a parallel theory exists for cubical subdivisions of cubical complexes, in which the role of the h-vector of a simplicial complex is played by the (short or long) cubical h-vector of a cubical complex, defined by Adin, and the role of the local h-vector of a triangulation of a simplex is played by the (short or long) cubical local h-vector of a cubical subdivision of a cube. The cubical local h-vectors are defined in this paper and are shown to share many of the properties of their simplicial counterparts. Generalizations to subdivisions of locally Eulerian posets are also discussed.     

Strongly Signable and Partitionable Posets

The class ofStrongly Signablepartially ordered sets is introduced and studied. It is show that strong signability, reminiscent of Björner–Wachs' recursive coatom orderability, provides a useful and broad sufficient condition for a poset to be dual CR and hence partitionable. The flagh-vectors of strongly signable posets are therefore non-negative. It is proved that recursively shellable posets, polyhedral fans, and face lattices of partitionable simplicial complexes are all strongly signable, and it is conjectured that all spherical posets are. It is concluded that the barycentric subdivision of a partitionable complex is again partitionable, and an algorithm for producing a partitioning of the subdivision from a partitioning of the complex is described. An expression for the flagh-polynomial of a simplicial complex in terms of itsh-vector is given, and is used to demonstrate that the flagh-vector is symmetric or non-negative whenever theh-vector is.     

Generic Cohen-Macaulay Monomial Ideals

Given a simplicial complex, it is easy to construct a generic deformation of its Stanley- Reisner ideal. The main question under investigation in this paper is how to characterize the simplicial complexes such that their Stanley-Reisner ideals have Cohen-Macaulay generic deformations. Algorithms are presented to construct such deformations for matroid complexes, shifted complexes, and tree complexes.AMS Subject Classification: 13P10, 13C14, 52B20, 52B40, 52B22.     

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.     

Resolutions of Stanley-Reisner rings and Alexander duality

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal I_Δ in the polynomial ring A = k[x₁, …, x_n], and its quotient k[Δ] = A/I_Δ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ^* which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim_k Tor_i^A (k[Δ],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ^*. As corollaries, we prove that I_Δ has a linear resolution as A-module if and only if Δ^* is Cohen-Macaulay over k, and show how to compute the Betti numbers dim_k Tor_i^A (k[Δ],k) in some cases where Δ^* is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.     

Higher commutators in the loop space homology of K-products

We consider a problem of calculating the loop space homology for so-called polyhedral products defined by an arbitrary simplicial complex K. A presentation of this homology algebra is obtained from the homology of the complements of diagonal subspace arrangements, which, in turn, is calculated using an infinite resolution of the exterior Stanley-Reisner algebra. We get an explicit presentation of the loop homology algebra for polyhedral products for classes of simplicial complexes such as flag complexes and the duals of sequentially Cohen-Macaulay complexes in terms of higher commutator products. We give a construction for the iteration of higher products and discuss the relationship between this problem and problems in commutative algebra.     

On toric face rings

Following a construction of Stanley we consider toric face rings associated to rational pointed fans. This class of rings is a common generalization of the concepts of Stanley-Reisner and affine monoid algebras. The main goal of this article is to unify parts of the theories of Stanley-Reisner and affine monoid algebras. We consider (non-pure) shellable fan’s and the Cohen-Macaulay property. Moreover, we study the local cohomology, the canonical module and the Gorenstein property of a toric face ring.     

Multicover inequalities on colored complexes

Suppose that C is a balanced simplicial complex. We show that its flag f-vector contains an interesting multiplicative structure. We define η _s (C):= log₂ f _S (C), and characterize the convex cone in which this flag η-vector may lie. Additionally, we specialize our results to the case when C is a pure balanced simplicial complex, and when C is a graded poset.     

A generalized Macaulay theorem and generalized face rings

We prove that the f-vector of members in a certain class of meet semi-lattices satisfies Macaulay inequalities 0?k^∂(fk)?fk−1 for all k?0. We construct a large family of meet semi-lattices belonging to this class, which includes all posets of multicomplexes, as well as meet semi-lattices with the “diamond property,” discussed by Wegner [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828], as special cases. Specializing the proof to the later family, one obtains the Kruskal-Katona inequalities and their proof as in [G. Wegner, Kruskal-Katona's theorem in generalized complexes, in: Finite and Infinite Sets, vol. 2, in: Colloq. Math. Soc. János Bolyai, vol. 37, North-Holland, Amsterdam, 1984, pp. 821-828].For geometric meet semi-lattices we construct an analogue of the exterior face ring, generalizing the classic construction for simplicial complexes. For a more general class, which also includes multicomplexes, we construct an analogue of the Stanley-Reisner ring. These two constructions provide algebraic counterparts (and thus also algebraic proofs) of Kruskal-Katona's and Macaulay's inequalities for these classes, respectively.     

Maximal Betti numbers of Cohen–Macaulay complexes with a given <Emphasis Type="Italic">f</Emphasis>-vector

Given the f-vector f = (f₀, f₁, . . .) of a Cohen–Macaulay simplicial complex, it will be proved that there exists a shellable simplicial complex Δ_f with f(Δ_f) = f such that, for any Cohen–Macaulay simplicial complex Δ with f(Δ) = f, one has for all i and j, where f(Δ) is the f-vector of Δ and where β _ij (I _Δ) are graded Betti numbers of the Stanley–Reisner ideal I _Δ of Δ. The first author is supported by JSPS Research Fellowships for Young Scientists. Received: 23 January 2006     

Cohen-Macaulay types of subgroup lattices of finite abelianp-groups

For a minimal free resolution of a Stanley-Reisner ring constructed from the order complex of a modular lattice. T. Hibi showed that its last Betti number (called the Cohen-Macaulay type) is computed by means of the Möbius function of the given modular lattice. Using this result, we consider the Stanley-Reisner ring of the subgroup lattice of a finite abelianp-group associated with a given partition, and show that its Cohen-Macaulay type is a polynomial inp with integer coefficients.