An Algebraic Introduction to K-Theory

This book is both an introduction to K-theory and a text in algebra. These two roles are entirely compatible. On the one hand, nothing more than the basic algebra of groups, rings, and modules is needed to explain the clasical algebraic K-theory. On the other hand, K-theory is a natural organizing principle for the standard topics of a second course in algebra, and these topics are presented carefully here. The reader will not only learn algebraic K-theory, but also Dedekind domains, class groups, semisimple rings, character theory, quadratic forms, tensor products, localization, completion, tensor algebras, symmetric algebras, exterior algebras, central simple algebras, and Brauer groups. The presentation is self-contained, with all the necessary background and proofs, and is divided into short sections with exercises to reinforce the ideas and suggest further lines of inquiry. The prerequisites are minimal: just a first semester of algebra (including Galois theory and modules over a principal ideal domain). No experience with homological algebra, analysis, geometry, number theory, or topology is assumed. The author has successfuly used this text to teach algebra to first year graduate students. Selected topics can be used to construct a variety of one-semester courses; coverage of the entire text requires a full year.

Algebraic K-theory is a discipline which is internally coherent, but has strong connections to diverse mathematical disciplines, and has contributed solutions to problems in algebra, number theory, analysis, geometry and functional analysis. It even has links to particle physics. This book serves as a text in graduate level algebra following a standard one semester algebra course. It also is an introduction to the field of algebraic K-theory and can serve as a reference to this subject for specialists in other parts of mathematics. Unlike other books on K-theory, no experience outside algebra is required of the reader.