Algebraic K-Theory

Introduction. I: Classical algebraic K-functors. 1. The Grothendieck functor Ko. 2. The Bass--Whitehead functor K£X/X£1. 3. The Milnor functor KM2. II: Higher K-functors. 1. K-Theory of Quillen for exact categories. 2. The Quillen plus construction. 3. K-Theory of Swan. 4. K-theory of Karoubi--Villamayor. 5. K-Theory of Waldhausen. III: Properties of algebraic K-functors. 1. Exactness, excision and the Mayor--Vietoris sequence. 2. The localization theorem. 3. The fundamental theorem. 4. Products in algebraic K-theory. 5. Stability. IV: Relations between algebraic K-theories. 1. Isomorphism of Quillen's algebraic K-theories. Agreement of plus construction and Q-construction. 2. Connection of Quillen's plus construction with Swan's algebraic K-theory. 3. Comparison of Swan's and Karoubi--Villamayor's algebraic K-theories. V: Relation between algebraic and topological K-theories. 1. Equivalence of categories of finitely generated projective modules and vector bundles over a compact space for C*-algebras. 2. K-theory of special normed algebras and Z2-graded C*-algebras. 3. Isomorphism of Swan's and Karoubi--Villamayor's K-theories with topological K-theory for real Banach algebras. VI: The Problem of Serre for polynomial and monoid algebras. 1. Proof ofAnderson's conjecture. 2. The algebraic proof of Swan. VII: Connection with cyclic homology. References. Index.