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Advanced Modern Algebra Third Edition, Part 2 GRADUATE STUDIES IN MATHEMATICS 180 Advanced Modern Algebra Third Edition, Part 2 Joseph J. Rotman American Mathematical Society 10.1090/gsm/180 GRADUATE STUDIES IN MATHEMATICS 180 Advanced Modern Algebra Third Edition, Part 2 Joseph J. Rotman American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky The 2002 edition of this book was previously published by Pearson Education, Inc. 2010 Mathematics Subject Classification. Primary 12-01, 13-01, 14-01, 15-01, 16-01, 18-01, 20-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-180 Library of Congress Cataloging-in-Publication Data Rotman, Joseph J., 1934– Advanced modern algebra / Joseph J. Rotman. – Third edition. volumes cm. – (Graduate studies in mathematics ; volume 165) Includes bibliographical references and index. ISBN 978-1-4704-1554-9 (alk. paper : pt. 1) ISBN 978-1-4704-2311-7 (alk. paper : pt. 2) 1. Algebra. I. Title. QA154.3.R68 2015 512–dc23 2015019659 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 select pages 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. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. Third edition, Part 2 c 2017 by the American Mathematical Society. All rights reserved. Third edition, Part 1 c 2015 by the American Mathematical Society. All rights reserved. Second edition c 2010 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 222120191817 Contents Foreword vii Preface to Third Edition: Part 2 ix Chapter C-1. More Groups 1 C-1.1. Group Actions 1 Graphs 16 Counting 20 C-1.2. Sylow Theorems 24 C-1.3. Solvable and Nilpotent Groups 33 Solvable Groups 34 Nilpotent Groups 43 C-1.4. Projective Unimodular Groups 50 General Linear Group GL(n, k)50 Simplicity of PSL(2,q)52 Simplicity of PSL(n, q)58 C-1.5. More Group Actions 66 Projective Geometry 67 Multiple Transitivity 74 PSL Redux 77 C-1.6. Free Groups and Presentations 81 Existence and Uniqueness of Free Groups 82 Presentations 92 C-1.7. Nielsen–Schreier Theorem 97 C-1.8. The Baer–Levi Proof 102 The Categories Simp and Simp∗ 102 Fundamental Group 104 Covering Complexes 110 Co-Galois Theory 115 iii iv Contents C-1.9. Free Products and the Kurosh Theorem 118 C-1.10. Epilog 124 Chapter C-2. Representation Theory 127 C-2.1. Artinian and Noetherian 127 C-2.2. Jacobson Radical 130 C-2.3. Group Actions on Modules 135 C-2.4. Semisimple Rings 137 C-2.5. Wedderburn–Artin Theorems 146 C-2.6. Introduction to Lie Algebras 161 C-2.7. Characters 168 C-2.8. Class Functions 176 C-2.9. Character Tables and Orthogonality Relations 180 C-2.10. Induced Characters 186 C-2.11. Algebraic Integers Interlude 193 C-2.12. Theorems of Burnside and of Frobenius 200 C-2.13. Division Algebras 208 Chapter C-3. Homology 223 C-3.1. Introduction 223 C-3.2. Semidirect Products 226 C-3.3. General Extensions and Cohomology 236 C-3.4. Complexes 255 C-3.5. Homology Functors 262 C-3.6. Derived Functors 271 C-3.7. Right Derived Functors 285 C-3.8. Ext and Tor 292 C-3.9. Cohomology of Groups 309 C-3.10. Crossed Products 326 C-3.11. Introduction to Spectral Sequences 333 Chapter C-4. More Categories 339 C-4.1. Additive Categories 339 C-4.2. Abelian Categories 344 C-4.3. g-Sheaves 359 C-4.4. Sheaves 368 C-4.5. Sheaf Cohomology 378 C-4.6. Module Categories 384 C-4.7. Adjoint Functor Theorem for Modules 392 Contents v C-4.8. Algebraic K-Theory 403 The Functor K0 404 The Functor G0 408 Chapter C-5. Commutative Rings III 419 C-5.1. Local and Global 419 Subgroups of Q 419 C-5.2. Localization 427 C-5.3. Dedekind Rings 445 Integrality 446 Algebraic Integers 455 Characterizations of Dedekind Rings 467 Finitely Generated Modules over Dedekind Rings 477 C-5.4. Homological Dimensions 486 C-5.5. Hilbert’s Theorem on Syzygies 496 C-5.6. Commutative Noetherian Rings 502 C-5.7. Regular Local Rings 510 Bibliography 527 Index 537 Foreword My friend and UIUC mathematics department colleague Joe Rotman was com- pletely dedicated to his series of books on algebra. He was correcting his draft of this revision of Advanced Modern Algebra during what sadly turned out to be his final hospital visit. At that time, Joe and his family asked me to do what I could to see this close-to-finished version to publication. Two more friends and colleague of Joe’s, Jerry Janusz and Paul Weichsel, joined the project. Jerry did a meticulous line-by-line reading of the manuscript, and all three of us answered questions posed by the AMS editorial staff, based on Arlene O’Sean’s very careful reading of the manuscript. It is clear that this book would have been even richer if Joe had been able to continue to work on it. For example, he planned a chapter on algebraic geome- try. We include the first paragraph of that chapter, an example of Joe’s distinctly personal writing style, as a small memorial to what might have been. Mathematical folklore is the “standard” mathematics “everyone” knows. For example, all mathematics graduate students today are familiar with elementary set theory. But folklore changes with time; elementary set theory was not part of nineteenth-century folklore. When we write a proof, we tacitly use folklore, usually not mentioning it explicitly. That folklore depends on the calendar must be one of the major factors complicating the history of math- ematics. We can find primary sources and read, say, publications of Ruffini at the beginning of the 1800s, but we can’t really follow his proofs unless we are familiar with his contemporary folklore. I want to express my thanks to Sergei Gelfand and Arlene O’Sean of the AMS and to Jerry Janusz and Paul Weichsel of UIUC for all their help. Our overrid- ing and mutual goal has been to produce a book which embodies Joe Rotman’s intentions. Bruce Reznick May 31, 2017 vii Preface to Third Edition: Part 2 The second half of this third edition of Advanced Modern Algebra hasPart1aspre- requisite. This is not to say that everything there must be completely mastered, but the reader should be familiar with what is there and should not be uncomfortable upon seeing the words category, functor, module,orZorn. The format of Part 2 is standard, but there are interactions between the dif- ferent chapters. For example, group extensions and factor sets are discussed in the chapter on groups as well as in the chapter on homology. I am reminded of my experience as an aspiring graduate student. In order to qualify for an advanced degree, we were required to take a battery of written exams, one in each of algebra, analysis, geometry, and topology. At the time, I felt that each exam was limited to its own area, but as I was wrestling with an algebra problem, the only way I could see to solve it was to use a compactness argument. I was uncomfortable: compactness arguments belong in the topology exam, not in the algebra exam! Of course, I was naive. The boundaries of areas dividing mathematics are artificial; they really don’t describe what is being studied but how it is being studied. It is a question of attitude and emphasis. Doesn’t every area of mathematics study polynomials? But algebraists and analysts view them from different perspectives. After all, mathematics really is one vast subject, and all its parts and emphases are related. A word about references in the text. If I mention Theorem C-1.2 or Exercise C-1.27 on page 19, then these are names in Part 2 of the third edition. References to names in Part 1 will have the prefix A- or B- and will say, for example, Theorem A-1.2 in Part 1 or Exercise B-1.27 on page 288 in Part 1. In an exercise set, an asterisk before an exercise, say, *C-1.26, means that this exercise is mentioned elsewhere in the text, usually in a proof. Thanks goes to Ilya Kapovich, Victoria Corkery, Vincenzo Acciaro, and Stephen Ullom. ix Bibliography 1.
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