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Lecture Notes in Mathematics 1747 Editors: A Lecture Notes in Mathematics 1747 Editors: A. Dold, Heidelberg F. Takens, Groningen B. Teissier, Paris Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo Lars Winther Christensen Gorenstein Dimensions ~ Springer Author Lars Winther Christensen Matematisk Afdeling Kcbenhavns Universitet Universitetsparken 5 2100 KObenhavns ~, Danmark E-maih [email protected] Cataloging-in-PublicationData applied for Die Deutsche Bibliothek - CIP-Einbeitsaufnahme: Winther Christensen,Lars: Gorensteindimensions / Lars Winther Christensen. - Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London ; Milan ; Pads ; Singapore ; Tokyo : Springer,2000 (Lecture notes in mathematics ; 1747) ISBN 3-540-41132-1 Mathematics Subject Classification (2000): 13-02, 13C 15, 13D02, 13D05, 13D07, 13D25, 13E05, 13HI0, 18G25 ISSN 0075-8434 ISBN 3-540-41132-1 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10724258 41/3142-543210 - Printed on acid-free paper Preface In 1995, almost five years ago, Hans-Bjcrn Foxby gave me a copy of Anneaux de Gorenstein, et torsion en alg~bre commutative, a set of notes based on lectures given by Auslander in 1966-67. I was told that the notes contained ideas about something called 'Gorenstein dimensions', a concept which had received renewed attention in the early 1990s and might prove to be an interesting topic for my Master's thesis. I was easily convinced: Gorenstein dimensions have been part of my life ever since. I have, already, expressed my gratitude to Foxby on several occasions, however, I wish to do it again: This book is an enhanced and extended version of my Master's thesis from 1996, and I thank Hans-Bjcrn Foxby for encouraging me to publish it and for his continual support during the entire process. Among the people who have helped me complete this project, my friend and colleague Srikanth Iyengar stands out. I was about to start the project when we first met in late 1998, and Srikanth has from day one shown a genuine and lasting interest in the project: reading at least one version of every chapter and making valuable comments on my style and significant improvements to several proofs. I also thank Anders Frankild and Mette Thrane Nielsen for reading parts of the manuscript, and Luchezar Avramov and Peter Jcrgensen for their readiness to discuss specific details in some of my proofs. Finally, I thank Line, my wife, for her endless love, support, and encourage- ment. The invaluable help and support from colleagues, friends, and family notwith- standing, this book is no better than its author. I have tried to set out the text in such a way that the main features stand out clearly, and I have taken great care to supply detailed proofs; it may sometimes seem that I go to great lengths to explain the obvious, but that is how I am. Copenhagen, June 2000 Lars Winther Christensen Contents Introduction 1 Synopsis 3 Conventions and Prerequisites 9 Notation and Basics ......................... 9 Standard Tools ............................ 11 Standard Homomorphisms ...................... 11 Homological Dimensions ....................... 13 A Hierarchy cf Rings ......................... 14 The Classical Gorenstein Dimension 17 1.1 The G-class .............................. 17 1.2 G-dimensi0n of Finite Modules ................... 22 1.3 Standard Operating Procedures ................... 29 1.4 Local Rings .............................. 32 1.5 G-dimension versus Projective Dimension ............. 37 G-dimension and Reflexive Complexes 41 2.1 Rei?exive Complexes ......................... 41 2.2 The Module Case ........................... 47 2.3 G-dimension cf Complexes with Finite Homology ......... 52 2.4 Testing G-dimension ......................... 58 Auslander Categories 65 3.1 The Auslander Class ......................... 65 3.2 The Bass Class ............................ 71 3.3 Foxby Equivalence .......................... 76 3.4 Cohen-Macaulay Rings ....................... 83 4 G-projectivity 91 4.1 The G-class Revisited ........................ 91 4.2 Gcrenstein Projective Modules ................... 97 4.3 G-projectives over Cohen-Macaulay Rings ............. 99 4.4 Gorenstein Projective Dimension .................. 105 vn"1"" CONTENTS G-flatness 113 5.1 Gorenstein Flat Modules ....................... 113 5.2 Gorenstein Flat Dimension ..................... 120 5.3 The Ultimate AB Formula ...................... 127 5.4 Comparing Tot-dimensions ..................... 131 G-injectivity 135 6.1 Gorenstein Injective Modules .................... 135 6.2 Gcrenstein Injective Dimension ................... 141 6.3 G-injective ~ersus G-flat Dimension ................ 148 6.4 Exercises in Stability ......................... 152 A Hyperhomology 159 A.1 Basic Definitions and Notation ................... 160 A.2 Standard Functors and Morphisms ................. 168 A.3 Resolutions .............................. 171 A.4 (Almost) Derived ~nctors ..................... 175 A.5 Homological Dimensions ....................... 180 A.6 Depth and Width ........................... 183 A.7 Numerical and Formal Invariants .................. 185 A.8 Dualizing Complexes ......................... 187 Bibliography 191 List of Symbols 197 Index 199 Introduction Introduction In 1967 Auslander [1] introduced a new invariant for finitely generated mo- dules over commutative Noetherian rings: a relative homological dimension called the G-dimension. The 'G' is, no doubt, for 'Gorenstein' and chosen because the following are equivalent for a local ring R: • R is Gorenstein. • The residue field Rim has finite G-dimension (m is the unique maximal ideal). • All finitely generated R-modules have finite G-dimension. This characterization of Gorenstein rings (rings of finite self-injective dimen- sion) is parallel to the Auslander-Buchsbaum-Serre characterization of regular rings (rings of finite global dimension), but to make the analogy complete a fourth condition, dealing with non-finitely generated modules, is needed. So far, the most successful approach to G-dimension for non-finitely gener- ated modules is the one taken in the 1990s by Enochs et al. in [22-32]. At first (quoting from the abstract of [32]) " ... to get good results it was necessary to take the base ring Gorenstein", but the theory of Foxby equivalence 1 has subsequently brought about good results over rings with dualizing complexes in general. In particular, Enochs' group [32] and Foxby [39] have outlined a beautiful theory for Gorenstein projective and fiat dimensions (extensions of the original G-dimension) and Gorenstein injective dimension (dual to the Goren- stein projective one) over Cohen-Macanlay local rings with a dualizing module. The purpose of this monograph is to give a detailed and up to date presentation of the theory of Gorenstein dimensions. In chapter 1 we review Auslander's G-dimension using homological algebra in the tradition of the fifties and sixties. In the second chapter we extend the G-dimension to complexes and start using hyperhomological algebra (an extension of homological algebra for modules). The Gorenstein projective, fiat, and injective dimensions are treated in chapters 4, 5, and 6, and the theory of Foxby equivalence is dealt with in chapter 3. The synopsis, following immediately after this introduction, gives an overview of the principal results. 1Some authors call it Foxby duality. 2 INTRODUCTION This book is intended as a reference for Gorenstein dimensions. It is aimed at mathematicians, especially graduate students, working with homological dimen- sions in commutative algebra. Indeed, any admirer of classics like the Auslander- Buchsbaum formula, the Auslander-Buchsbaum-Serre characterization of reg- ular rings, and Bass' formula for injective dimension must be intrigued by the highlights of this monograph. The reader is expected to be well-versed in commutative algebra and in the standard applications of homological methods within this realm. In chapters 2-6 we work consistently with complexes of modules, but for the benefit of those who prefer plain modules, all major results are restated for modules in traditional notation. The appendix offers a crash course in hyperhomological algebra, including homological dimensions. Hopefully, this easy reference will make the proofs accessible, also for casual users of hyperhomological methods. We work with categories because the language is convenient, but, apart from the basic definitions, no knowledge of category theory is required. To the relief of some -- and to the dismay of others -- it should be emphasized that we do not use the derived category: we use equivalence of complexes, but we never formally invert the quasi-isomorphisms. This deficiency does not really
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