Algebraic K-Theory plays an important role in many areas of modern mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. The broad range of these topics has tended to give the subject an aura of inapproachability. This book, based on a course at the University of Maryland in the fall of 1990, is intended to enable graduate students or mathematicians working in other areas not only to learn the basics of algebraic K-Theory, but also to get a feel for its many applications. The required prerequisites are only the standard one-year graduate algebra course and the standard introductory graduate course on algebraic and geometric topology. Many topics from algebraic topology, homological algebra, and algebraic number theory are developed as needed. The final chapter gives a concise introduction to cyclic homology and its interrelationship with K-Theory.
Author(s): Jonathan Rosenberg
Series: Graduate Texts in Mathematics 147
Edition: 1 (corr. second printing)
Publisher: Springer
Year: 1994
Language: English
Commentary: Corrected second printing, 1996
Pages: 404
Tags: K-Theory; Algebraic K-Theory
Preface v
Chapter 1. K_0 of Rings 1
1. Defining K_0 1
2. K_0 from idempotents 7
3. K_0 of PIDs and local rings 11
4. K_0 of Dedekind domains 16
5. Relative K_0 and excision 27
6. An application: Swan's Theorem and topological K-theory 32
7. Another application: Euler characteristics and the Wall finiteness obstruction 41
Chapter 2. K_1 of Rings 59
1. Defining K1 59
2. K_1 of division rings and local rings 62
3. K_1 of PIDs and Dedekind domains 74
4. Whitehead groups and Whitehead torsion 83
5. Relative K_1 and the exact sequence 92
Chapter 3. K_0 and K_1 of Categories, Negative K -Theory 108
1. K_0 and K_1 of categories, G_0 and G_1 of rings 108
2. The Grothendieck and Bass-HeIler-Swan Theorems 132
3. Negative K-theory 153
Chapter 4. Milnor's K_2 162
1. Universal central extensions and H_2 162
Universal central extensions 163
Homology of groups 168
2. The Steinberg group 187
3. Milnor's K_2 199
4. Applications of K_2 218
Computing certain relative K_1 groups 218
K_2 of fields and number theory 221
Almost commuting operators 237
Pseudo-isotopy 240
Chapter 5. The +-Construction and Quillen K-Theory 245
1. An introduction to classifying spaces 245
2. Quillen's +-construction and its basic properties 265
3. A survey of higher K-theory 279
Products 279
K-theory of fields and of rings of integers 281
The Q-construction and results proved with it 289
Applications 295
Chapter 6. Cyclic homology and its relation to K-THeory 302
1. Basics of cyclic homology 302
Hochschild homology 302
Cyclic homology 306
Connections with "non-commutative de Rham theory" 325
2. The Chern character 331
The classical Chern character 332
The Chern character on K_0 335
The Chern character on higher K-theory 340
3. Some applications 350
Non-vanishing of class groups and Whitehead groups 350
Idempotents in C*-algebras 355
Group rings and assembly maps 362
References 369
Index 383
