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\> # * * »/ where dim A = d 1 and exactly ft faces of A have i + 1 elements. Define for positive integers m9