Algebraic K-Theory and Its Applications

Algebraic K-Theory plays an important role in many areas of mathematics: most notably algebraic topology, number theory, and algebraic geometry, but even including operator theory. This book is intended to enable graduate students or mathematicians working in other areas to learn the basics of algebraic K-Theory and to get a feel for its applications. Exercises are an integral part of the book.

Inhalt

1. K0 of Rings.- 1. Defining K0.- 2. K0 from idempotents.- 3. K0 of PIDs and local rings.- 4. K0 of Dedekind domains.- 5. Relative K0 and excision.- 6. An application: Swan's Theorem and topological K- theory.- 7. Another application: Euler characteristics and the Wall finiteness obstruction.- 2.K1 of Rings.- 1. Defining K1.- 2. K1 of division rings and local rings.- 3. 1 of PIDs and Dedekind domains.- 4. Whitehead groups and Whitehead torsion.- 5. Relative K1 and the exact sequence.- 3. K0 and K1 of Categories, Negative K-Theory.- 1. K0 and K1 of categories, Go and G1 of rings.- 2. The Grothendieck and Bass-Heller-Swan Theorems.- 3. Negative K-theory.- 4. Milnor's K2.- 1. Universal central extensions and H2.- 2. The Steinberg group.- 3. Milnor's K2.- 4. Applications of K2.- 5. The +?Construction and Quillen K-Theory.- 1. An introduction to classifying spaces.- 2. Quillen's +?construction and its basic properties.- 3. A survey of higher K-theory.- 6. Cyclic homology and its relation to K-Theory.- 1. Basics of cyclic homology.- 2. The Chern character.- 3. Some applications.- References.- Books and Monographs on Related Areas of Algebra, Analysis, Number Theory, and Topology.- Books and Monographs on Algebraic K-Theory.- Specialized References.- Notational Index.