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 K0 and K1 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. Cover S Title The K-Book: An Introduction to Algebraic K-theory Copyright © 2013 by Charles A. Weibel ISBN 978-0-8218-9132-2 QA612.33.W45 2013 512'.66-dc23 LCCN 2012039660 Contents Preface Acknowledgements Chapter I Projective modules and vector bundles 1. Free modules, GLn, and stably free modules EXERCISES 2. Projective modules EXERCISES 3. The Picard group of a commutative ring EXERCISES 4. Topological vector bundles and Chern classes EXERCISES 5. Algebraic vector bundles EXERCISES Chapter II The Grothendieck group Ko 1. The group completion of a monoid EXERCISES 2. Ko of a ring EXERCISES 3. K(X), KO(X), and KU(X) of a topological space EXERCISES 4. Lambda and Adams operations EXERCISES 5. Ko of a symmetric monoidal category EXERCISES 6. Ko of an abelian category EXERCISES 7. Ko of an exact category EXERCISES 8. Ko of schemes and varieties EXERCISES 9. KO of a Waldhausen category EXERCISES Appendix. Localizing by calculus of fractions EXERCISES Chapter III K1 and K2 of a ring 1. The Whitehead group K1 of a ring EXERCISES 2. Relative K1 EXERCISES 3. The Fundamental Theorems for K1 and Ko EXERCISES 4. Negative K-theory EXERCISES 5. K2 of a ring EXERCISES 6. K2 of fields EXERCISES 7. Milnor K-theory of fields EXERCISES Chapter IV Definitions of higher K-theory 1. The BGL+ definition for rings EXERCISES 2. K-theory with finite coefficients EXERCISES 3. Geometric realization of a small category EXERCISES 4. Symmetric monoidal categories EXERCISES 5. $\lambda$-operations in higher K-theory EXERCISES 6. Quillen's Q-construction for exact categories EXERCISES 7. The "+ = Q" Theorem EXERCISES 8. Waldhausen's wS. construction EXERCISES 9. The Gillet-Grayson construction EXERCISES 10. Nonconnective spectra in K-theory EXERCISES 11. Karoubi-Villamayor K-theory EXERCISES 12. Homotopy K-theory EXERCISES Chapter V The Fundamental Theoremsof higher K-theory 1. The Additivity Theorem EXERCISES 2. Waldhausen localization and approximation EXERCISES 3. The Resolution Theorems and transfer maps EXERCISES 4. Devissage EXERCISES 5. The Localization Theorem for abelian categories EXERCISES 6. Applications of the Localization Theorem EXERCISES 7. Localization for K_ (R) and K_(X) EXERCISES 8. The Fundamental Theorem for K* (R) and K(X) EXERCISES 9. The coniveau spectral sequence of Gersten and Quillen EXERCISES 10. Descent and Mayer-Vietoris properties EXERCISES 11. Chern classes EXERCISES Chapter VI The higher K-theory of fields 1. K-theory of algebraically closed fields EXERCISES 2. The e-invariant of a field EXERCISES 3. The K-theory of R EXERCISES 4. Relation to motivic cohomology EXERCISES 5. K3 of a field EXERCISES 6. Global fields of finite characteristic EXERCISES 7. Local fields EXERCISES 8. Number fields at primes where cd = 2 EXERCISES 9. Real number fields at the prime 2 EXERCISES 10. The K-theory of Z EXERCISES Bibliography Index of notation Index Back Cover 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.