xvi PREFACE that our guiding principles are quite different from Yoshino’s, and con- sequently there are few topics on which our presentation parallels that in [Yos90]. When it does, it is generally because both books follow the original presentation of Auslander, Auslander-Reiten, or Yoshino. Early versions of this book have been used for advanced graduate courses at the University of Nebraska in Fall 2007 and at Syracuse University in Fall 2010. In each case, the students had had at least one full-semester course in commutative algebra at the level of Mat- sumura’s book [Mat89]. A few more advanced topics are needed from time to time, such as the basics of group representations and character theory, properties of canonical modules and Gorenstein rings, Cohen’s structure theory for complete local rings, the Artin-Rees Lemma, and the material on multiplicity and Serre’s conditions in the Appendix. Many of these can be taken on faith at first encounter, or covered as extra topics. The core of the book, Chapters 3 through 9, is already more ma- terial than could comfortably be covered in a semester course. One remedy would be to streamline the material, restricting to the case of complete local rings with algebraically closed residue fields of character- istic zero. One might also skip or sketch some of the more tangential material. We regard the following as essential: Chapter 3 (omitting most of the proof of Theorem 3.7) the first three sections of Chap- ter 4 Chapter 5 Chapter 6 (omitting the proof of Theorem 6.11, the calculations in §3, and §4) Chapters 7 and 8 and the first two sections of Chapter 9. Chapters 2 and 10 can each stand alone as optional top- ics, while the thread beginning with Chapters 11 and 13, continuing through Chapters 15 and 17 could serve as the basis of a completely separate course (though some knowledge of the first half of the book would be necessary to make sense of Chapters 14 and 16). At the end of each chapter is a short section of exercises of varying diﬃculty, over 120 in all. Some are independent problems, while others ask the solver to fill in details of proofs omitted from the body of the text. We gratefully acknowledge the many, many people and organiza- tions whose support we enjoyed while writing this book. Our students at Nebraska and Syracuse endured early drafts of the text, and helped

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