Combinatorial decompositions of rings and almost Cohen-Macaulay complexes
The author and \textit{A. M. Garsia} have proved in an earlier paper [Adv. Math. 39, 155-184 (1981; Zbl 0466.13003)] that a finitely generated graded algebra R over a field K always has what they call a ''combinatorial decomposition''. Such a decomposition means a homogeneous system of parameters \(\theta_ 1,...,\theta_ r\) of R together with a lift S of a basis of \(R/(\theta_ 1,...,\theta_ r)\) such that a basis of R is obtained in a suitable manner in terms of S and certain monomials in \(\theta_ 1,...,\theta_ r\). In the present paper the author shows how to find S for a given system \(\theta_ 1,...,\theta_ r\). He then describes explicitly the Hilbert series and a combinatorial decomposition for the Stanley-Reisner ring K[\(\Delta]\) associated to a completely balanced almost Cohen-Macaulay finite simplicial complex \(\Delta\). Also, a ring-theoretic expression is given for the homology groups of such a \(\Delta\).
scientific article; zbMATH DE number 3855258