Advanced Modern Algebra, Part 2

570 Pages·2017·10.232 MB·English

by  Joseph J. Rotman

GRADUATE STUDIES 180
IN MATHEMATICS
Advanced
Modern Algebra
Third Edition, Part 2
Joseph J. Rotman
American Mathematical Society
GRADUATE STUDIES 180
IN MATHEMATICS
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 PearsonEducation, Inc.
2010 Mathematics Subject Classification. Primary 12-01, 13-01, 14-01, 15-01, 16-01,
18-01, 20-01.
For additional informationand updates on this book, visit
www.ams.org/bookpages/gsm-180
Library of Congress Cataloging-in-Publication Data
Rotman,JosephJ.,1934–
Advancedmodernalgebra/JosephJ.Rotman. –Thirdedition.
volumescm. –(Graduatestudiesinmathematics;volume165)
Includesbibliographicalreferencesandindex.
ISBN978-1-4704-1554-9(alk.paper: pt.1)
ISBN978-1-4704-2311-7(alk.paper: pt.2)
1.Algebra. I.Title.
QA154.3.R68 2015
512–dc23 2015019659
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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 K 404
0
The Functor G 408
0
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.
TwomorefriendsandcolleagueofJoe’s,JerryJanuszandPaulWeichsel,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
notmentioningitexplicitly. Thatfolkloredependsonthecalendar
mustbeoneofthemajorfactorscomplicatingthehistoryofmath-
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
ThesecondhalfofthisthirdeditionofAdvanced ModernAlgebra hasPart1aspre-
requisite. Thisisnottosaythateverythingtheremustbecompletelymastered,but
the reader should be familiar with what is there and should not be uncomfortable
upon seeing the words category, functor, module, or Zorn.
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,wewererequiredtotakeabatteryofwrittenexams, oneineachofalgebra,
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.
Afterall,mathematicsreallyisonevastsubject, andallitspartsandemphasesare
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
tonamesinPart1willhavetheprefixA-orB-andwillsay, forexample, 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.
ThanksgoestoIlyaKapovich,VictoriaCorkery,VincenzoAcciaro,andStephen
Ullom.
ix