``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.

  • Introduction
  • Chapter I: Projective Modules and Vector Bundles (53pp.)

  • Chapter II: The Grothendieck group K_0 (104 pp.)

  • Chapter III: K_1 and K_2 of a ring (70 pp.)

  • Chapter IV: Definitions of higher K-theory (92 pp.)

  • Chapter V: The Fundamental Theorems of higher K-theory (87 pp)

  • Chapter VI: The higher K-theory of Fields (66 pp)

  • References

    This book grew like Topsy!

    Back Story:

    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 Algebraic K-theory, 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!)

    Errata 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