Let~ be a finite simplicial complex (or complex for short) on the vertex set V={Xl,"" Xn}. Thus,~ is a collection of subsets of V satisfying the two conditions:(i){xi} E~ for all xi E V, and (ii) if Fe~ and GCF, then Ge~. There is a certain commutative ring A~ which is closely associated with the combinatorial and topological properties of~. We will discuss this association in the special case when A~ is a Cohen-Macaulay ring. Lack of space prevents us from giving most of the proofs and from commenting on a number of interesting sidelights. However, a greatly expanded version of this paper is being planned. Let~ be a complex (= finite simplicial complex). If Fe~, we call F a face of~. If F has i+ 1 elements (denoted card F= i+ 1),~ ay dim F= i. Let d= 0+ 1= max {card FIF E~}. We write dim~= 0= d-1. If every maximal face of~ has dimension 0, then~ is called pure (or homogeneous by topologists). Let fi be the number of i-dimensional faces of~. Thus fo= n. The vector f=(fo, f1,•.•, fo) is called the f-vector of~. Now define a function on the non-negative integers by