Generalized Cohen-Macaulay dimension.
type: Item
instance of: scholarly article
title: Generalized Cohen-Macaulay dimension.
zbMATH Open document ID: 1077.13005
DOI: 10.1016/S0021-8693(03)00333-8
published in: Journal of Algebra
publication date: 2004-03-14T00:00:00Z
review text: 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.
reviewed by: Cornel Baetica
Mathematics Subject Classification ID: 13C14
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 2055674
zbMATH Keywords: Generalized Cohen-Macaulay ring
zbMATH Keywords: Gorenstein dimension
zbMATH Keywords: generalized Cohen-Macaulay dimension
zbMATH Keywords: GCM-dimension
zbMATH Keywords: Auslander-Buchsbaum depth
zbMATH Keywords: flat extensions
zbMATH Keywords: quasi-Buchsbaum dimension
zbMATH Keywords: almost Cohen-Macaulay dimension
author: Javad Asadollahi
author: Shokrollah Salarian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2055674