Synopses & Reviews
Algebraic K-theory is a modern branch of algebra which has many important applications in fundamental areas of mathematics connected with algebra, topology, algebraic geometry, functional analysis and algebraic number theory. Methods of algebraic K-theory are actively used in algebra and related fields, achieving interesting results.
This book presents the elements of algebraic K-theory, based essentially on the fundamental works of Milnor, Swan, Bass, Quillen, Karoubi, Gersten, Loday and Waldhausen. It includes all principal algebraic K-theories, connections with topological K-theory and cyclic homology, applications to the theory of monoid and polynomial algebras and in the theory of normed algebras.
This volume will be of interest to graduate students and research mathematicians who want to learn more about K-theory.
Synopsis
Algebraic K-theory is a modern branch of algebra which has many important applications in fundamental areas of mathematics connected with algebra, topology, algebraic geometry, functional analysis and algebraic number theory. Methods of algebraic K-theory are actively used in algebra and related fields, achieving interesting results. This book presents the elements of algebraic K-theory, based essentially on the fundamental works of Milnor, Swan, Bass, Quillen, Karoubi, Gersten, Loday and Waldhausen. It includes all principal algebraic K-theories, connections with topological K-theory and cyclic homology, applications to the theory of monoid and polynomial algebras and in the theory of normed algebras. This volume will be of interest to graduate students and research mathematicians who want to learn more about K-theory.
Table of Contents
Introduction.
I: Classical algebraic K-functors. 1. The Grothendieck functor
K_{o}. 2. The Bass--Whitehead functor
K£X/X£_{1}. 3. The Milnor functor
K^{M}_{2}. 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 Z
_{2}-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 of Anderson's conjecture.
2. The algebraic proof of Swan.
VII: Connection with cyclic homology. References. Index.