Description
Springer Algebraic K-Theory and Its Applications by Jonathan Rosenberg
Algebraic K-Theory is crucial in many areas of modern mathematics, especially algebraic topology, number theory, algebraic geometry, and operator theory. This text is designed to help graduate students in other areas learn the basics of K-Theory and get a feel for its many applications. Topics include algebraic topology, homological algebra, algebraic number theory, and an introduction to cyclic homology and its interrelationship with K-Theory._x000D_ Table of contents :- _x000D_
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.- Universal central extensions.- Homology of groups.- 2. The Steinberg group.- 3. Milnor's K2.- 4. Applications of K2.- Computing certain relative K1 groups.- K2 of fields and number theory.- Almost commuting operators.- Pseudo-isotopy.- 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.- Products.- K-theory of fields and of rings of integers.- The Q-construction and results proved with it.- Applications.- 6. Cyclic homology and its relation to K-Theory.- 1. Basics of cyclic homology.- Hochschild homology.- Cyclic homology.- Connections with "non-commutative de Rham theory".- 2. The Chern character.- The classical Chern character.- The Chern character on K0.- The Chern character on higher K-theory.- 3. Some applications.- Non-vanishing of class groups and Whitehead groups.- Idempotents in C*-algebras.- Group rings and assembly maps.- 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._x000D_