The K-Book: An Introduction to Algebraic K-theory (Graduate Studies in Mathematics, 145)

Informally, $K$-theory is a tool for probing the structure of a mathematical object such as a ring or a topological space in terms of suitably parameterized vector spaces and producing important intrinsic invariants which are useful in the study of algebraic and geometric questions. Algebraic $K$-theory, which is the main character of this book, deals mainly with studying the structure of rings. However, it turns out that even working in a purely algebraic context, one requires techniques from homotopy theory to construct the higher $K$-groups and to perform computations. The resulting interplay of algebra, geometry, and topology in $K$-theory provides a fascinating glimpse of the unity of mathematics. This book is a comprehensive introduction to the subject of algebraic $K$-theory. It blends classical algebraic techniques for $K_0$ and $K_1$ with newer topological techniques for higher $K$-theory such as homotopy theory, spectra, and cohomological descent. The book takes the reader from the basics of the subject to the state of the art, including the calculation of the higher $K$-theory of number fields and the relation to the Riemann zeta function. Algebraic K-theory is a field of abstract algebra concerning projective modules over a ring and vector bundles over schemes. It has many applications in mathematics such as algebraic topology and geometry, number theory and operator theory. In advanced physics there are applications such as string theory, D-brane theory and condensed matter physics. In this graduate level textbook Weibel presents the reader with a detailed overview of the field. Topics include Chern classes, Picard groups, Algebraic vector bundles, basic as well as more complex constructions of K0 and K1, as well as higher order K-groups Kn. There are many exercises at the end of each chapter and rigorous proofs throughout. Annotation 2013 Book News, Inc., Portland, OR (booknews.com) Projective Models And Vector Bundles -- The Grothendieck Group K0 -- K1and K2 Of A Ring -- Definitions Of Higher K-theory -- Fundamental Theorms Of Higher K-theory. Charles A. Weibel. Includes Bibliographical References And Index.