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**Gorenstein dimensions.**
*(English)*
Zbl 0965.13010

Lecture Notes in Mathematics. 1747. Berlin: Springer. viii, 204 p. (2000).

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 Gorenstein flat dimension the author uses the restricted Tor-dimension introduced by Foxby. Chapter 6 (“G-injectivity”) introduces Gorenstein injective modules and Gorenstein injective dimension. In this chapter is also studied the duality between G-flatness and G-injectivity. Especially strong results are described in this book over Cohen-Macaulay local rings.

The book is very well written; new concepts and results are compared with the classical ones, and ample motivations are provided. For the benefit of the reader an appendix on hyperhomological algebra (homological algebra for complexes rather than for modules) is included. The book is useful for both research and teaching: it can be used for graduate courses and seminars.

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 Gorenstein flat dimension the author uses the restricted Tor-dimension introduced by Foxby. Chapter 6 (“G-injectivity”) introduces Gorenstein injective modules and Gorenstein injective dimension. In this chapter is also studied the duality between G-flatness and G-injectivity. Especially strong results are described in this book over Cohen-Macaulay local rings.

The book is very well written; new concepts and results are compared with the classical ones, and ample motivations are provided. For the benefit of the reader an appendix on hyperhomological algebra (homological algebra for complexes rather than for modules) is included. The book is useful for both research and teaching: it can be used for graduate courses and seminars.

Reviewer: M.Roitman (Haifa)

### MSC:

13D05 | Homological dimension and commutative rings |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

13D02 | Syzygies, resolutions, complexes and commutative rings |

18G10 | Resolutions; derived functors (category-theoretic aspects) |

13D07 | Homological functors on modules of commutative rings (Tor, Ext, etc.) |

13D25 | Complexes (MSC2000) |

13E05 | Commutative Noetherian rings and modules |

13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |

18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |