Advanced Modern Algebra: Third Edition, Part 2 (Graduate Studies in Mathematics)

This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten.The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively.The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.Abstract: This is the second part of the new edition of Advanced Modern Algebra. The material has been significantly reorganised and sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra. Title Page 2 Copyright 3 Table of Contents 4 Foreword 8 Preface to Third Edition: Part 2 10 C-1. More Groups 12 C-1.1. Group Actions 12 C-1.2. Sylow Theorems 35 C-1.3. Solvable and Nilpotent Groups 44 C-1.4. Projective Unimodular Groups 61 C-1.5. More Group Actions 77 C-1.6. Free Groups and Presentations 92 C-1.7. Nielsen鈥揝chreier Theorem 108 C-1.8. The Baer鈥揕evi Proof 113 C-1.9. Free Products and the Kurosh Theorem 129 C-1.10. Epilog 135 C-2. Representation Theory 138 C-2.1. Artinian and Noetherian 138 C-2.2. Jacobson Radical 141 C-2.3. Group Actions on Modules 146 C-2.4. Semisimple Rings 148 C-2.5. Wedderburn鈥揂rtin Theorems 157 C-2.6. Introduction to Lie Algebras 172 C-2.7. Characters 179 C-2.8. Class Functions 187 C-2.9. Character Tables and Orthogonality Relations 191 C-2.10. Induced Characters 197 C-2.11. Algebraic Integers Interlude 204 C-2.12. Theorems of Burnside and of Frobenius 211 C-2.13. Division Algebras 219 C-3. Homology 234 C-3.1. Introduction 234 C-3.2. Semidirect Products 237 C-3.3. General Extensions and Cohomology 247 C-3.4. Complexes 266 C-3.5. Homology Functors 273 C-3.6. Derived Functors 282 C-3.7. Right Derived Functors 296 C-3.8. Ext and Tor 303 C-3.9. Cohomology of Groups 320 C-3.10. Crossed Products 337 C-3.11. Introduction to Spectral Sequences 344 C-4. More Categories 350 C-4.1. Additive Categories 350 C-4.2. Abelian Categories 355 C-4.3. g-Sheaves 370 C-4.4. Sheaves 379 C-4.5. Sheaf Cohomology 389 C-4.6. Module Categories 395 C-4.7. Adjoint Functor Theorem for Modules 403 C-4.8. Algebraic K-Theory 414 C-5. Commutative Rings III 430 C-5.1. Local and Global 430 C-5.2. Localization 438 C-5.3. Dedekind Rings 456 C-5.4. Homological Dimensions 497 C-5.5. Hilbert鈥檚 Theorem on Syzygies 507 C-5.6. Commutative Noetherian Rings 513 C-5.7. Regular Local Rings 521 Bibliography 538 Index 548 This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study. This new edition, now in two parts, has been significantly reorganized and many sections have been rewritten. This first part, designed for a first year of graduate algebra, consists of two courses: Galois theory and Module theory.