Combinatorial Decompositions of Rings and Almost Cohen-Macaulay Complexes

The concept of a combinatorial decomposition of a graded K algebra was introduced by Baclawski-Garsia [4], and they showed that every (finitelygenerated) graded K algebra has such a decomposition. The purpose of this paper is to prove some general properties of combinatorial decompositions, which are useful for finding such decompositions. We then show how to compute combinatorial decompositions for a class of rings based on simplicial complexes. This class of rings is utilized in the theory of lexicographic rings. ([ 31). Another interesting consequence of our investigation (Section 4) is a ring-theoretic interpretation of the homology groups of a triangulated compact manifold. Throughout the paper we use N for the semigroup of nonnegative integers and K to denote a field; and, unless specified otherwise, cohomology will always be computed with coefficients in K. Moreover, most rings will be finitely generated N’-graded K algebras for some 1. For such a ring R, we write R, or XsR for the graded part of multidegree S E N’. We will think of S as a multisubset of [I] = (l,..., 1). The Hilbert series of R is the power series H(R; t) = H(R; t, ,..., t,) = &ER\I, dim,(zsR) tS, where tS is the (multiset) product JJ IES ti. The Krull dimension of R is the order of the pole at t = 1 of the power series H(R; t,..., t). Given two power series F(t) and G(t) in the same variables t ,,..., t,, we write F(t) < G(t) to mean that a, < b, for every S E N’, where F(t) = C a, tS and G(t) = 2 6, tS. Given two homogeneous ideals I and J of R, the ideal quotient is the homogeneous ideal (I: J) = (f E R 1 fI E J}, A related concept is the annihilator of a homogeneous ideal Z in a graded R-module N, given by an%(O= isENI&=(O)l ([ll).