Combinatorial decompositions of rings and almost Cohen-Macaulay complexes
type: Item
instance of: scholarly article
title: Combinatorial decompositions of rings and almost Cohen-Macaulay complexes
zbMATH Open document ID: 0538.13008
DOI: 10.1016/0021-8693(83)90132-1
author: Kenneth Baclawski
published in: Journal of Algebra
publication date: 1983-01-01T00:00:00Z
review text: 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 \(\Delta\).
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 3855258
zbMATH Keywords: almost Cohen-Macaulay complexes
zbMATH Keywords: combinatorial decomposition
zbMATH Keywords: system of parameters
zbMATH Keywords: Hilbert series
zbMATH Keywords: Stanley-Reisner ring
reviewed by: Balwant Singh
MaRDI profile type: Publication
full work available at URL: 0021-8693(83)90132-1
OpenAlex ID: W2001003135
cites work: Q5563439
cites work: Cohen-Macaulay ordered sets
cites work: Rings with lexicographic straightening law
cites work: Combinatorial decompositions of a class of rings
cites work: On the number of faces of simplicial complexes and the purity of Frobenius
cites work: Q3265726
cites work: The Upper Bound Conjecture and Cohen-Macaulay Rings
cites work: Balanced Cohen-Macaulay Complexes
description: scientific article; zbMATH DE number 3855258