Algebraic K-theory of a Finite Field

The main goal of the present project is to give a sketch of the calculation of the algebraic K-theory of a finite field. Not all the details are developed, but several references are given where detailed proofs may be found. However, the main source of the project is D.J. Benson’s book on ”Representations and Cohomolgy: Cohomolgy of groups and modules” [4]. Section 1 is an introduction to principal G-bundles and uses them in order to define classifying spaces. Special attention is given to the fiber bundles coming from the Stiefel and Grassmann manifolds. These are later used to define the space BU as the union of complex Grassmann manifolds of with the weak topology. Section 2 goes further and uses BU to give a general definition of topological K-theory for paracompact spaces. Certain fundamental properties of topological K-theory are presented in this section as Bott periodicity and Adam’s operations. Section 3 gives a brief introduction to homotopy fixed points and how to give to this space and additive structure. Furthermore, it is shown that one can think of the Adam’s operations ψ as self-maps of BU, and the space Fψ is defined as the homotopy fixed points of such self-maps. Moreover, it is shown that Fψ is the fiber of a give fibration BU → BU and thus its homotopy groups are easily calculated from the long exact sequence in homotopy. At the end of the section, it is shown that one can construct a map BGL(Fq) → Fψ well defined up to homotopy, that induces a homology isomorphism. Section 4 defines Quillen’s plus construction and his definition of the algebraic K-groups for an arbitrary ring. Finally, Section 5 blends the results of all the previous sections in order to calculate the algebraic K-groups of a finite field by means of the map BGL(Fq)→ Fψ constructed previously.