Generalized Cohen–Macaulay dimension

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