Gorenstein dimensions
type: Item
instance of: scholarly article
title: Gorenstein dimensions
zbMATH Open document ID: 0965.13010
author: Lars Winther Christensen
published in: Lecture Notes in Mathematics
publication date: 2000-12-14T00:00:00Z
review text: This book deals with Gorenstein dimensions of Noetherian rings. The Gorenstein dimension was defined and used by Auslander to characterize Gorenstein local rings. Properties of the Gorenstein dimension are parallelized in the book under review to those of the projective dimension, and in this context the Auslander class is introduced. Extending the theory to non-finite modules yields the concept of Gorenstein projective dimension. Also are studied Gorenstein flat dimension and Gorenstein injective dimension. I quote from the author's synopsis: ``As indicated in this short outline, the monograph focuses on the Gorenstein dimensions ability to characterize Gorenstein rings; and the coverage of interrelations between Gorenstein dimensions takes clues from the domestic triangle: projective-injective-flat dimension''. We now briefly review the contents of each chapter. In chapter 1 (``The classical Gorenstein dimension'') the G-class for finite modules is defined and studied. The G-dimension of finite modules is defined using resolutions in the G-class and is compared with the projective dimension. Chapter 2 extends the G-dimension to complexes with finite homology; as indicated by its title (``G-dimension and reflexive complexes''), reflexive complexes play here a crucial role. I quote from chapter 3 (``Auslander categories''): ``For a local ring \(R\) with a dualizing complex we will introduce \(\dots\) two full subcategories of \(R\)-complexes: the Auslander class and the Bass class. They are -- together with the full subcategory of reflexive complexes -- known as Auslander categories, and they are linked together by Foxby equivalence.'' Chapter 4 (``G-projectivity'') extends the theory of chapter 1 to non-finite modules using Gorenstein projective modules, which were introduced by Enochs and Jenda. Chapter 5 (``G-flatness'') deals with ``Gorenstein flat modules: a notion which includes both usually flat modules and Gorenstein projective modules''. In the study of the…
Mathematics Subject Classification ID: 13D05
Mathematics Subject Classification ID: 13-02
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 18G10
Mathematics Subject Classification ID: 13D07
Mathematics Subject Classification ID: 13D25
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 18G25
zbMATH DE Number: 1544059
zbMATH Keywords: Auslander class
zbMATH Keywords: Bass class
zbMATH Keywords: Cohen-Macaulay local rings
zbMATH Keywords: derived category
zbMATH Keywords: dualizing complex
zbMATH Keywords: Foxby duality
zbMATH Keywords: Gorenstein dimension
zbMATH Keywords: Gorenstein projective dimension
zbMATH Keywords: Gorenstein injective dimension
zbMATH Keywords: Gorenstein flat dimension
zbMATH Keywords: hyperhomology
zbMATH Keywords: reflexive complex
reviewed by: Moshe Roitman
MaRDI profile type: Publication
DOI: 10.1007/BFB0103980
description: scientific article; zbMATH DE number 1544059