Cohen-Macaulay Complexes

Let Δ be a finite simplicial complex (or complex for short) on the vertex set V = (x1,…,xn). Thus, Δ is a collection of subsets of V satisfying the two conditions: (i) (xi) e Δ for all xi e V, and (ii) if F e Δ and G ⊂ F, then G e Δ. There is a certain commutative ring AΔ which is closely associated with the combinatorial and topological properties of Δ. We will discuss this association in the special case when AΔ 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.