Cohen-Macaulay Representations

Mathematical Surveys and Monographs Volume 181 Cohen-Macaulay Representations Graham J. Leuschke Roger Wiegand American Mathematical Society http://dx.doi.org/10.1090/surv/181 Cohen-Macaulay Representations Mathematical Surveys and Monographs Volume 181 Cohen-Macaulay Representations Graham J. Leuschke Roger Wiegand American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair MichaelA.Singer Jordan S. Ellenberg Benjamin Sudakov MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 13C14, 16G50; Secondary 13B40, 13C05, 13C60, 13H10, 16G10, 16G60, 16G70. For additional information and updates on this book, visit www.ams.org/bookpages/surv-181 Library of Congress Cataloging-in-Publication Data Leuschke, Graham J., 1973– Cohen-Macaulay representations / Graham J. Leuschke, Roger Wiegand. p. cm. — (Mathematical surveys and monographs ; v. 181) Includes bibliographical references and index. ISBN 978-0-8218-7581-0 (alk. paper) 1. Cohen-Macaulay modules. 2. Representations of rings (Algebra) I. Wiegand, Roger, 1943– II. Title. QA251.3.L48 2012 512.44—dc23 2012002344 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 171615141312 To Conor, David and Andrea, and to our students, past, present and future. Contents Preface xi Chapter 1. The Krull-Remak-Schmidt Theorem 1 §1. KRS in an additive category 1 §2. KRS over Henselian rings 6 §3. R-modules vs. R-modules 7 §4. Exercises 9 Chapter 2. Semigroups of Modules 13 §1. Krull monoids 14 §2. Realization in dimension one 17 §3. Realization in dimension two 23 §4. Flat local homomorphisms 26 §5. Exercises 27 Chapter 3. Dimension Zero 29 §1. Artinian rings with finite Cohen-Macaulay type 29 §2. Artinian pairs 32 §3. Exercises 39 Chapter 4. Dimension One 41 §1. Necessity of the Drozd-Ro˘ıter conditions 42 §2. Sufficiency of the Drozd-Ro˘ıter conditions 45 §3. ADE singularities 50 §4. The analytically ramified case 52 §5. Multiplicity two 54 §6. Ranks of indecomposable MCM modules 56 §7. Why MCM modules? 57 §8. Exercises 58 Chapter 5. Invariant Theory 61 §1. The skew group ring 61 §2. The endomorphism algebra 65 vii viii CONTENTS §3. Group representations and the McKay-Gabriel quiver 71 §4. Exercises 77 Chapter 6. Kleinian Singularities and Finite CM Type 81 §1. Invariant rings in dimension two 81 §2. Kleinian singularities 83 §3. McKay-Gabriel quivers of the Kleinian singularities 91 §4. Geometric McKay correspondence 97 §5. Exercises 106 Chapter 7. Isolated Singularities and Dimension Two 109 §1. Miyata’s theorem 109 §2. Isolated singularities 113 §3. Two-dimensional CM rings of finite CM type 117 §4. Exercises 120 Chapter 8. The Double Branched Cover 123 §1. Matrix factorizations 123 §2. The double branched cover 128 §3. Knörrer’s periodicity 133 §4. Exercises 139 Chapter 9. Hypersurfaces with Finite CM Type 141 §1. Simple singularities 141 §2. Hypersurfaces in good characteristics 144 §3. Gorenstein rings of finite CM type 150 §4. Matrix factorizations for the Kleinian singularities 151 §5. Bad characteristics 159 §6. Exercises 160 Chapter 10. Ascent and Descent 163 §1. Descent 163 §2. Ascent to the completion 164 §3. Ascent along separable field extensions 170 §4. Equicharacteristic Gorenstein singularities 172 §5. Exercises 173 Chapter 11. Auslander-Buchweitz Theory 175 §1. Canonical modules 175 §2. MCM approximations and FID hulls 179 §3. Numerical invariants 189 CONTENTS ix §4. The index and applications to finite CM type 194 §5. Exercises 200 Chapter 12. Totally Reflexive Modules 203 §1. Stable Hom and Auslander transpose 203 §2. Complete resolutions 207 §3. Totally reflexive modules 209 §4. Exercises 215 Chapter 13. Auslander-Reiten Theory 217 §1. AR sequences 217 §2. AR quivers 224 §3. Examples 228 §4. Exercises 238 Chapter 14. Countable Cohen-Macaulay Type 241 §1. Structure 241 §2. Burban-Drozd triples 244 §3. Hypersurfaces of countable CM type 252 §4. Other examples 260 §5. Exercises 263 Chapter 15. The Brauer-Thrall Conjectures 267 §1. The Harada-Sai lemma 268 §2. Faithful systems of parameters 270 §3. Proof of Brauer-Thrall I 276 §4. Brauer-Thrall II 280 §5. Exercises 285 Chapter 16. Finite CM Type in Higher Dimensions 287 §1. Two examples 287 §2. Classification for homogeneous CM rings 293 §3. Exercises 295 Chapter 17. Bounded CM Type 297 §1. Hypersurface rings 297 §2. Dimension one 299 §3. Descent in dimension one 303 §4. Exercises 307 Appendix A. Basics and Background 309 xCONTENTS §1. Depth, syzygies, and Serre’s conditions 309 §2. Multiplicity and rank 313 §3. Henselian rings 317 Appendix B. Ramification Theory 321 §1. Unramified homomorphisms 321 §2. Purity of the branch locus 326 §3. Galois extensions 335 Bibliography 341 Index 355 Preface This book is about the representation theory of commutative local rings, specifically the study of maximal Cohen-Macaulay modules over Cohen-Macaulay local rings. The guiding principle of representation theory, broadly speaking, is that we can understand an algebraic structure by studying the sets upon which it acts. Classically, this meant understanding finite groups by studying the vector spaces they act upon; the powerful tools of lin- ear algebra can then be brought to bear, revealing information about the group that was otherwise hidden. In other branches of representation theory, such as the study of finite-dimensional associative algebras, sophisticated technical machinery has been built to investigate the properties of modules, and how restrictions on modules over a ring restrict the structure of the ring. The representation theory of maximal Cohen-Macaulay modules began in the late 1970s and grew quickly, inspired by three other ar- eas of algebra. Spectacular successes in the representation theory of finite-dimensional algebras during the 1960s and 70s set the standard for what one might hope for from a representation theory. In particular, this period saw: P. Gabriel’s introduction of the representations of quivers and his theorem that a quiver has finite representation type if and only if it is a disjoint union of ADE Coxeter-Dynkin diagrams; M. Auslander’s influential Queen Mary notes applying his work on functor categories to representation theory; Auslander and I. Reiten’s foundational work on AR sequences; and key insights from the Kiev school, particularly Y. Drozd, L. A. Nazarova, and A. V. Ro˘ıter. All these advances continued the work on finite representation type be- gun in the 1940s and 50s by T. Nakayama, R. Brauer, R. Thrall, and J. P. Jans. Secondly, the study of lattices over orders, a part of integral representation theory, blossomed in the late 1960s. Restricting atten- tion to lattices rather than arbitrary modules allowed a rich theory to xi xii PREFACE develop. In particular, the work of Drozd-Ro˘ıter and H. Jacobinski around this time introduced the conditions we call “the Drozd-Ro˘ıter conditions” classifying commutative orders with only a finite number of non-isomorphic indecomposable lattices. Finally, M. Hochster’s study of the homological conjectures emphasized the importance of the maximal Cohen-Macaulay condition (even for non-finitely generated modules). The equality of the geometric invariant of dimension with the arithmetic one of depth makes this class of modules easy to work with, simultaneously ensuring that they faithfully reflect the structure of the ring. The main focus of this book is on the problem of classifying Cohen- Macaulay local rings having only a finite number of indecomposable maximal Cohen-Macaulay modules, that is, having finite CM type. No- tice that we wrote “the problem,” rather than “the solution.” Indeed, there is no complete classification to date. There are many partial results, however, including complete classifications in dimensions zero and one, a characterization in dimension two under some mild assump- tions, and a complete understanding of the hypersurface singularities with this property. The tools developed to obtain these classifications have many applications to other problems as well, in addition to their inherent beauty. In particular there are applications to the study of other representation types, including countable type and bounded type. This is not the first book about the representation theory of Cohen- Macaulay modules over Cohen-Macaulay local rings. The text [Yos90] by Y. Yoshino is a fantastic book and an invaluable resource, and has inspired us both on countless occasions. It has been the canonical ref- erence for the subject for twenty years. In those years, however, there have been many advances. To give just two examples, we mention C. Huneke and Leuschke’s elementary proof in 2002 of Auslander’s theorem that finite CM type implies isolated singularity, and R. Wiegand’s 2000 verification of F.-O. Schreyer’s conjecture that finite CM type as- cends to and descends from the completion. These developments alone might justify a new exposition. Furthermore, there are many facets of the subject not covered in Yoshino’s book, some of which we are quali- fied to describe.

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