The K-book: an introduction to algebraic K-theory

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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


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