Cohen-Macaulay Simplicial Complexes Research Articles

Relative Cohen-Macaulayness and relative unmixedness of bigraded modules

In this paper we study the finitely generated bigraded modules over a standard bigraded polynomial ring which are relative Cohen-Macaulay or relative unmixed with respect to one of the irrelevant bigraded ideals. A generalization of Reisner's criterion for Cohen-Macaulay simplicial complexes is considered.

Pure and Cohen–Macaulay Simplicial Complexes Associated with Squarefree Lexsegment Ideals

We classify the unmixed squarefree lexsegment ideals and determine those which are Cohen–Macaulay.

Face numbers of generalized balanced Cohen-Macaulay complexes

A common generalization of two theorems on the face numbers of Cohen-Macaulay (CM, for short) simplicial complexes is established: the first is the theorem of Stanley (necessity) and Björner-Frankl-Stanley (sufficiency) that characterizes all possible face numbers of a-balanced CM complexes, while the second is the theorem of Novik (necessity) and Browder (sufficiency) that characterizes the face numbers of CM subcomplexes of the join of the boundaries of simplices.

On Face Vectors of Barycentric Subdivisions of Manifolds

We study face vectors of barycentric subdivisions of simplicial homology manifolds. Recently, Kubitzke and Nevo proved that the g-vector of the barycentric subdivision of a Cohen–Macaulay simplicial complex is an M-vector, which in particular proves the g-conjecture for barycentric subdivisions of simplicial homology spheres. In this paper, we prove an analogue of this result for Buchsbaum simplicial posets and simplicial homology manifolds.

(Bi-)Cohen–Macaulay Simplicial Complexes and Their Associated Coherent Sheaves#

ABSTRACT Via the BGG correspondence, a simplicial complex Δ on [n] is transformed into a complex of coherent sheaves on P n−1. We show that this complex reduces to a coherent sheaf ℱ exactly when the Alexander dual Δ* is Cohen–Macaulay. We then determine when both Δ and Δ* are Cohen–Macaulay. This corresponds to ℱ being a locally Cohen–Macaulay sheaf. Lastly, we conjecture for which range of invariants of such Δ's it must be a cone, and show the existence of such Δ's which are not cones outside of this range.

On face numbers of manifolds with symmetry

Necessary conditions on the face numbers of Cohen–Macaulay simplicial complexes admitting a proper action of the cyclic group Z/p Z of a prime order are given. This result is extended further to necessary conditions on the face numbers and the Betti numbers of Buchsbaum simplicial complexes with a proper Z/p Z -action. Adin's upper bounds on the face numbers of Cohen–Macaulay complexes with symmetry are shown to hold for all ( d−1)-dimensional Buchsbaum complexes with symmetry on n⩾3 d−2 vertices. A generalization of Kühnel's conjecture on the Euler characteristic of 2 k-dimensional manifolds and Sparla's analog of this conjecture for centrally symmetric 2 k-manifolds are verified for all 2 k-manifolds on n⩾6 k+3 vertices. Connections to the Upper Bound Theorem are discussed and its new version for centrally symmetric manifolds is established.

Algebraic Shifting and Sequentially Cohen-Macaulay Simplicial Complexes

Björner and Wachs generalized the definition of shellability by dropping the assumption of purity; they also introduced the $h$-triangle, a doubly-indexed generalization of the $h$-vector which is combinatorially significant for nonpure shellable complexes. Stanley subsequently defined a nonpure simplicial complex to be sequentially Cohen-Macaulay if it satisfies algebraic conditions that generalize the Cohen-Macaulay conditions for pure complexes, so that a nonpure shellable complex is sequentially Cohen-Macaulay. We show that algebraic shifting preserves the $h$-triangle of a simplicial complex $K$ if and only if $K$ is sequentially Cohen-Macaulay. This generalizes a result of Kalai's for the pure case. Immediate consequences include that nonpure shellable complexes and sequentially Cohen-Macaulay complexes have the same set of possible $h$-triangles.

On the Stanley ring of cubical complex

We investigate the properties of the Stanley ring of a cubical complex, a cubical analogue of the Stanley-Reisner ring of a simplicial complex. We compute its Hilbert series in terms of thef-vector, and prove that by taking the initial ideal of the defining relations, with respect to the reverse lexicographic order, we obtain the defining relations of the Stanley-Reisner ring of the triangulation via “pulling the vertices” of the cubical complex. Applying an old idea of Hochster we see that this ring is Cohen-Macaulay when the complex is shellable, and we show that its defining ideal is generated by quadrics when the complex is also a subcomplex of the boundary complex of a convex cubical polytope. We present a cubical analogue of balanced Cohen-Macaulay simplicial complexes: the class of edge-orientable shellable cubical complexes. Using Stanley's results about balanced Cohen-Macaulay simplicial complexes and the degree two homogeneous generating system of the defining ideal, we obtain an infinite set of examples for a conjecture of Eisenbud, Green, and Harris. This conjecture says that theh-vector of a polynomial ring inn variables modulo an ideal which has ann-element homogeneous system of parameters of degree two, is thef-vector of a simplicial complex.

A Monotonicity Property of h-vectors and h*-vectors

If Δ is a Cohen-Macaulay simplicial complex of dimension d - 1 and Δ′ is a Cohen-Macaulay subcomplex of dimension e - 1, such that no e + 1 vertices of Δ′ form a face of Δ, then we show that h (Δ′) ⩽ h(Δ), where h denotes the h-vector. In particular, h(Δ′) ⩽ h(Δ) if Δ and Δ′ are Cohen-Macaulay of the same dimension. Using similar techniques we obtain a class of Gorenstein complexes Δ, the h-vector of which is unimodal. Most of these results were obtained earlier by Kalai in a somewhat more complicated way. We then use our methods to give an analogous monotonicity property of Ehrhart polynomials of lattice polytopes (and more general objects). Our results on Ehrhart polynomials may be regarded as 'lattice analogues' of the well known monotonicity results concerning intrinsic volumes or quermassintegrals.