``The K-book: An introduction to algebraic K-theory''

1. Free and stably free modules; p.1
2. Projective modules; p.6
3. The Picard group of a ring; p.15
4. Topological vector bundles and Chern classes; p.26
5. Algebraic vector bundles. p.38

1. group completion of a monoid; p.1
2. K_0 of a ring; p.5
3. K(X) of a topological space; p.17
4. Lambda and Adams operations; p.24
5. K_0 of a symmetric monoidal category; p.37
6. K_0 of an abelian category; p.45
7. K_0 of an exact category; p.59
8. K_0 of schemes and varieties; p.73
9. K_0 of a Waldhausen category; p.83
Appendix: localizing by categories of fractions. p.97

1. K_1 of a ring; p.1
2. Relative K_1; p.13
3. the Fundamental Theorems for K_1 and K_0; p.17
4. Negative K-theory; p.27
5. K_2 of a ring; p.33
6. K_2 of fields; p.45
7. Milnor K-theory of fields. p.58

1. The BGL+ definition for Rings p.2
2. Geometric realization of a small category; p.15
3. Group completions for symmetric monoidal categories; p.27
4. λ-operations in higher K-theory p.38
5. Quillen's Q-construction for exact categories; p.43
6. The ``+=Q'' theorem; p.48
7. Waldhausen's wS. construction; p.54
8. The Gillet-Grayson construction; p.64
9. Non-connective spectra for algebraic $K$-theory p.67
10 Karoubi-Villamayor K-theory; p.70
11 Homotopy K-theory; p.77
12 K-theory with finite coefficients; p.81

1. The Additivity theorem; p. 1
2. Changes in Waldhausen Structure; p. 7
3. The Resolution Theorem; p.18
4. Devissage; p.24
5. Localization Sequences for G_*; p.26
6. Applications of the Localization Theorem p.28
7. Localization sequences for K(R) p.33

Watch it grow like Topsy!

Suggestions are welcome!

• Chapter VI: Higher K-theory of Fields
  (Here's the Trieste Lecture VIII on this)
  Promised Hilbert 90 for $K_2$, Merkurjev-Suslin result (III 6.9.3, 6.10.4), Brauer-Severi trick (III 7.8.2)
  K of finite fields & algebraically closed fields, Browder's calculation modulo p, K₃F

• Chapter VII: Applications to Algebraic Geometry

• Chapter VIII: Higher Chow Groups and Motivic Cohomology
  (I refer you to the book "Lectures in Motivic Cohomology" (Δ^• is defined in IV 9.3)
  Here are the Trieste Lecture IX (higher Chow groups) and Lecture X (motivic cohomology) on this.

• References .

Back Story:

It actually took a decade before the rumor became true... After a few years, when I had heard the rumor from at least a dozen people, I talked to Hy Bass, the author of the classic book Algebraic K-theory, about what would be involved in writing such a K-book. It was scary, because (in 1988) I didn't know even how to write a book. I needed a warm-up exercise, a practice book if you will. The result, An introduction to homological algebra, took over five years to write.

By this time (1995), the K-theory landscape had changed, and with it my vision of what my K-book should be. Was it an obsolete idea? After all, the new developments in Motivic Cohomology were affecting our knowledge of the K-theory of fields and varieties. In addition, there was no easily accessible source for this new material. In 1999, I was asked to turn a series of lectures by Voevodsky into a book. This project took over six years, in collaboration with Carlo Mazza and Volodia Voevodsky. The result was the book Lecture Notes on Motivic Cohomology.

Will this be it?

Thanks for corrections go to:

Errata for Jon Rosenberg's 1994 book on K-theory