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The $$K$$-book: An Introduction to Algebraic $$K$$-theory
Charles A. Weibel, Rutgers University, New Brunswick, NJ
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2013; 618 pp; hardcover
Volume: 145
ISBN-10: 0-8218-9132-4
ISBN-13: 978-0-8218-9132-2
List Price: US$89 Member Price: US$71.20
Order Code: GSM/145

Lecture Notes on Motivic Cohomology - Carlo Mazza, Vladimir Voevodsky and Charles Weibel

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.

Graduate students and research mathematicians interested in number theory, homological algebra, and $$K$$-theory.

Reviews

"Charles Weibel's 'K-book' offers a plethora of material from both classical and more recent algebraic K-theory. It is a perfect source book for seasoned graduate students and working researchers who are willing and eager to follow the author's expository path and who are ready for a lot of additional reading and self-reliant work. The many instructive examples and clarifying remarks help the reader grasp the essentials of algebraic K-theory from a panoramic view, and the entire exposition represents a highly valuable and useful guide to the subject in all its diversity and topicality. Although barely being a textbook or neophyte in the field, despite the wealth of background material sketched wherever necessary, the book, under review, is certainly the most topical presentation of algebraic K-theory at this time and an excellent enhancement of the existing literature in any case."

-- Newsletter of the European Mathematical Society

"Weibel presents his important and elegant subject with the authority of an experienced insider, placing stresses where they should be, presenting motivations and characterizations (always succinctly) so as to familiarize the reader with the shape of the subject. ... it contains a great number of examples, woven beautifully into the narrative, and excellent exercises."

-- MAA Reviews