``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.74
9. K_0 of a Waldhausen category; p.85
Appendix: localizing by categories of fractions. p.100

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. K-theory with finite coefficients; p.17
3. Geometric realization of a small category; p.23
4. Symmetric monoidal categories; p.35
5. λ-operations in higher K-theory; p.46
6. Quillen's Q-construction for exact categories; p.52
7. The ``+=Q'' theorem; p.60
8. Waldhausen's wS. construction; p.65
9. The Gillet-Grayson construction; p.75
10 Non-connective spectra for algebraic $K$-theory; p.78
11 Karoubi-Villamayor K-theory; p.81
12 Homotopy K-theory; p.88

1. The Additivity theorem; p. 1
2. Waldhausen localization and Approximation p. 11
3. The Resolution theorems and transfer maps; p.19
4. Devissage; p.32
5. The Localization Theorem for abelian categories; p.34
6. Applications of the Localization Theorem; p.37
7. Localization for K_*(R) and K_*(X); p.50
8. The Fundamental Theorem for K_*(R) and K_*(X); p.58
9. The coniveau spectral sequence; p.62
10 Mayer-Vietoris properties; p.70
11 Chern classes; p.77

1. K-theory of algebraically closed fields; p.1
2. The e-invariant of a field; p.6
3. the K-theory of R; p.11
4. Relation to motivic cohomology; p.15
5. K_3 of a field; p.22
6. Global fields of finite characteristic; p.35
7. Local fields; p.40
8. Number fields at primes where cd=2; p.45
9. Real number fields at the prime 2; p.48
10 The K-theory of Z; p.58

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" for this material.)
  

• References .

Back Story:

It actually took a decade before the rumor became true... In 1988 I wrote out a brief outline, following Quillen's paper Higher algebraic K-theory I. It was overwhelming. 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. Nevertheless, I wrote early versions of Chapters I-IV during 1994-1999.

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 Vladimir Voevodsky. The result was the book Lecture Notes on Motivic Cohomology, published in 2006.

In 2004-2008, Chapters IV and V were completed. At the same time, the final steps in the proof of the Norm Residue Theorem were finished. (This settles not just the Bloch-Kato Conjecture, but also the Beilinson-Lichtenbaum Conjectures and Quillen-Lichtenbaum Conjectures.) The proof of this theorem is scattered over a dozen papers and preprints, and writing it spanned over a decade of work, mostly by Rost and Voevodsky. Didn't it make sense to put this house in order? It did. I am currently collaborating with Christian Haesemeyer in writing a self-contained proof of this theorem.

Will this be it?

Thanks for corrections go to:

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