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Algebraic K-theory of Schemes

  • Marek Szyjewski
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

The present notes contain the foundations of algebraic K-theory to- gether with a series of explicit computations of the K-groups of fields and some classical varieties. Our goal is to provide an introduction to a more advanced reading, as well as to convince the reader that such a study may be useful and interesting. The exposition is by no means complete nor self-contained. We hope nevertheless, that the covered part of the theory is sufficient for effective computations in algebraic geometry.

The organization of these notes follows the historical development of algebraic K-theory in the 2nd half of the XXth century. We give a brief outline of the theory of the Grothendieck groups Ko(X), Ko(X). We also develop the higher K-theory (of Milnor and Quillen) of fields and compute the K-groups of finite fields. Next, the Quillen’s definition of K-groups as homotopy groups is given and their properties are discussed. Some instructive examples are also included. Moreover, we compute the higher algebraic K-groups of projective bundles, Brauer-Severi varieties and quadrics. In the end we apply these techniques to compute the Chow ring of a split quadric and to prove “Hilbert 90” for K2 (F( \(\sqrt a \) )) following Merkurjev’s proof.

Keywords

Exact Sequence Vector Bundle Abelian Category Coherent Sheave Central Simple Algebra 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Marek Szyjewski
    • 1
  1. 1.Instytut Matematyki Uniwersytetu ŚlaskiegoKatowicePoland

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