Algebraic K-Theory and Its Applications

ISBN-13: 9780387942483 / Angielski / Twarda / 1994 / 394 str.

J. Rosenberg; Jonathan Rosenberg

Opis

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.

Spis treści

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.