Generalized Cohen-Macaulay dimension.
For finitely generated modules over a commutative local Noetherian ring, the authors define a new homological dimension, called generalized Cohen-Macaulay dimension (GCM-dimension, for short), which characterizes the generalized Cohen-Macaulay rings in the same way the projective dimension characterizes the regular rings, that is a ring \(R\) is generalized Cohen-Macaulay if and only if every finitely generated \(R\)-module has finite GCM-dimension. We mention that some properties of GCM-dimension are similar to those of the Cohen-Macaulay dimension. For example, when the GCM-dimension of a module is finite, an Auslander-Buchsbaum depth formula holds for the localization of the module at every non-maximal prime, and moreover it equals the Cohen-Macaulay dimension of the localized module. In fact, an Auslander-Buchsbaum depth formula holds if and only if the GCM-dimension equals the Cohen-Macaulay dimension and it is finite. It is also shown that the GCM-dimension behaves as well as any other homological dimension with respect to flat extensions. At the end, the authors introduce a quasi-Buchsbaum dimension (that fits between GCM-dimension and Cohen-Macaulay dimension), and an almost Cohen-Macaulay dimension (that fits between restricted flat dimension and Cohen-Macaulay dimension) for any finitely generated module over a commutative local Noetherian ring, and they point out that these are really homological dimensions with the expected properties.
scientific article; zbMATH DE number 2055674