## 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 *K*_{o}(*X*), *K*′_{o}(*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 *K*_{2} (*F*(
\(\sqrt a \)
)) following Merkurjev’s proof.

## Keywords

Exact Sequence Vector Bundle Abelian Category Coherent Sheave Central Simple Algebra## Preview

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