**Graduate Studies in Mathematics**

Volume: 145;
2013;
618 pp;
Hardcover

MSC: Primary 19;

**Print ISBN: 978-0-8218-9132-2
Product Code: GSM/145**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-0943-2
Product Code: GSM/145.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

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#### Supplemental Materials

# The \(K\)-book: An Introduction to Algebraic \(K\)-theory

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*Charles A. 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.

#### Readership

Graduate students and research mathematicians interested in number theory, homological algebra, and \(K\)-theory.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## The $K$-book: An Introduction to Algebraic $K$-theory

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Preface ix10 free
- Projective modules and vector bundles 114 free
- The Grothendieck group 𝐾₀ 6982
- 𝐾₁ and 𝐾₂ of a ring 197210
- Definitions of higher 𝐾-theory 283296
- The Fundamental Theorems of higher 𝐾-theory 401414
- The higher 𝐾-theory of fields 509522
- Bibliography 589602
- Index of notation 599612 free
- Index 605618 free
- Back Cover Back Cover1633