K-theory of finite fields

Remarks

The theorem is due to Quillen. He used Brauer lifting to build a map from $BGL(\F_q)$ to a space $F\Psi^q$. This space is the homotopy fibre of the map $\psi^q - 1$ on the space $BU$ of topological K-theory, where $\psi^q$ is an Adams operation. He computed the homology of the groups $GL_n(\F_q)$ and showed that this map induces an isomorphism in homology, so that $BGL(\F_q)^+ \simeq F\Psi^q$, the plus construction of $BGL(\F_q)$. The homotopy groups of $F\Psi^q$ come from the long exact sequence of the fibration and Bott periodicity: $\psi^q$ acts on $\pi_{2i}(BU) \cong \Z$ by multiplication by $q^i$.

Check in low degrees: $K_1(\F_q) = \F_q^\times$ is cyclic of order $q - 1$, as the formula says for $i = 1$. Also $K_2(\F_q) = 0$, which agrees with the classical fact that every Steinberg symbol over a finite field is trivial. For $q = 2$ the formula gives $K_3(\F_2) \cong \Z/3$.

Going deeper

For a student

The computation is a model for the whole subject: Quillen's plus construction turns a problem about the homology of the groups $GL_n(\F_q)$ into a homotopy computation. It feeds into many later results, for example the localization sequence for rings of integers, whose terms include the K-groups of the residue fields. It also gives the K-theory of the algebraic closure of $\F_p$, by passing to the colimit over the fields $\F_{p^m}$.