Generalized Cohen-Macaulay dimension

Abstract

A new homological dimension, called GCM-dimension, will be defined for any finitely generated module M over a local Noetherian ring R. GCM-dimension (short for Generalized Cohen-Macaulay dimension) characterizes Generalized Cohen-Macaulay rings in the sense that: a ring R is Generalized Cohen-Macaulay if and only if every finitely generated R-module has finite GCM-dimension. This dimension is finer than CM-dimension and we have equality if CM-dimension is finite. Our results will show that this dimension has expected basic properties parallel to those of the homological dimensions. In particular, it satisfies an analog of the Auslander-Buchsbaum formula. Similar methods will be used for introducing quasi-Buchsbaum and Almost Cohen-Macaulay dimensions, which reflect corresponding properties of their underlying rings. © 2004 Elsevier Inc. All rights reserved.