``The K-book: an introduction to algebraic K-theory''
Errata to the published version of the K-book.
Note: the page numbers below are for the individual chapters,
and differ from the page numbers in the published version of
the K-book. The Theorem/Definition/Exercise numbers are the same.
Chapter I: Projective Modules and Vector Bundles
1. Free and stably free modules; p.1
Chapter II: The Grothendieck group K_0
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
Chapter III: K_1 and K_2 of a ring (70 pp.)
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 calculus of fractions. p.100
1. K_1 of a ring; p.1
Chapter IV: Definitions of higher K-theory (92 pp.)
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
Chapter V: The Fundamental Theorems of higher K-theory
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
Chapter VI: The higher K-theory of Fields
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
References (this is a .dvi file)
2. The e-invariant of a field; p.6
3. the K-theory of R; p.12
4. Relation to motivic cohomology; p.15
5. K_3 of a field; p.23
6. Global fields of finite characteristic; p.37
7. Local fields; p.42
8. Number fields at primes where cd=2; p.47
9. Real number fields at the prime 2; p.50
10 The K-theory of Z; p.60
This book grew like Topsy!
In 1985, I started hearing a persistent rumor that I was writing a book
on algebraic K-theory. This was a complete surprise to me!
After a few years, I had heard the rumor from at least a dozen people,
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
about what would be involved in writing such a 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-theory 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.
The project became known as the ``K-book'' at this time.
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. After discussions
with Voevodsky, I began collaborating with Christian Haesemeyer
in writing a self-contained proof of this theorem; this was published in
2019 as The Norm Residue Theorem in Motivic Cohomology.
Thanks for corrections go to:
R. Thomason, M. Lorenz, J. Csirik, M. Paluch, T. Geisser, Paul Smith,
P.A. Ostvaer, D. Grayson, I. Leary, A. Heider, P. Polo, J. Hornbostel,
B. Calmes, G. Garkusha, P. Landweber, A. Fernandez Boix, J.-L. Loday,
J. Davis, C. Crissman, R. Brasca, O. Braeunling, F. Calegari, K. Kedlaya,
D. Grinberg, P. Boavida, R. Reis, J. Levikov, O.Schnuerer, P.Pelaez,
Sujatha, J.Spakula, J.Cranch, A.Asok, G.Wilson, M.Szymik, I.Coley,
K. Paranjape, M. Lindh,
(your name can go here!)
for Jon Rosenberg's 1994 book on K-theory
Topsy is a character in Harriet B. Stowe's 1852 book
Uncle Tom's Cabin who claimed to have never been born:
``Never was born...
I 'spect I grow'd. Don't think nobody never made me.'' (sic)
Partially supported by many NSF and NSA grants over the decades