Cohen-Macaulay rings and constructible polytopes

We wish to point out how certain concepts in commutative algebra are of value in studying combinatorial properties of simplicial complexes. In particular, we obtain new restrictions on the /-vectors of simplicial convex polytopes. Let A be a finite simplicial complex with vertex set V = vl, v2, vn. We call the elements of A the faces of A. If the largest face of A has d elements, then we say dim A = d- 1. The f-vector of A is (fo f\\ # * * »/*-i) where dim A = d- 1 and exactly ft faces of A have i + 1 elements. Define for positive integers m9 d-1 H (A, m) =j: fifr-A Also define H (A, 0) = 1. We say that A is constructible 2 if it can be obtained by the following recursive procedure: (a) Every simplex is constructible, and (b) if A and A are constructible of the same dimension d, and if A n A is constructible of dimension d- 1, then A U A is constructible. We know of two main classes of constructible As: (A) The boundary complex of a simplicial convex polytope is constructible. This follows from 1. (B) Let D be a finite distributive lattice, and let D be D with the top element and bottom element removed. Let A be the simplicial complex whose faces are the chains of D. Then A is constructible. If h and i are positive integers, then h can be written uniquely in the form