On the Stanley ring of cubical complex

Abstract

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.