The algebraic K-theory of finite fields

Hi all, and apologies for the delay in posts. I was participating in the Intel Science Talent Search, which was in Washington, DC. In some of my questions, I was asked to compute the algebraic K-theory of finite fields, and I wanted to give an outline of this awesome computation.

Let $\mathbf{Z}/q\mathbf{Z}$ denote the finite field with $q$ elements. We’ll outline the computation of $K_i(\mathbf{Z}/q\mathbf{Z})$, but in order to do that, we need some homotopy theory. The Adams operations on $KU$ are given by the adams operations $\psi^k:\mathrm{BU}\to \mathrm{BU}$, in the sense that the Adams operations on $KU$ are given by $KU(X) = \mathrm{Map}(X,\mathrm{BU})\xrightarrow{\mathrm{Map}(X,\psi^k)}\mathrm{Map}(X,\mathrm{BU}) = KU(X)$. Quillen proved the following theorem:

Theorem (Quillen): The space $\mathrm{BGL}(\mathbf{Z}/q\mathbf{Z})^+$ is the homotopy fiber of $\mathrm{BU}\xrightarrow{\psi^q-1} \mathrm{BU}$.

How does this help us? Well, $\psi^q$ is just multiplication by $p^k$ on $\pi_{2k}BU$, because $\pi_{2k}BU$ is simply $KU(S^{2k})$. Consequently, we get the following computation of the K-groups of $\mathbf{Z}/q\mathbf{Z}$:

The K-groups of $\mathbf{Z}/q\mathbf{Z}$: for $n$ even, $K_n(\mathbf{Z}/q\mathbf{Z}) = 0$. For $n = 2m - 1$ odd, $K_n(\mathbf{Z}/q\mathbf{Z})\cong \mathbf{Z}/(q^m - 1)$. (Quillen also showed that if $\mathbf{Z}/q\mathbf{Z}\subseteq\mathbf{F}_{q^\prime}$ then the induced map $K_n(\mathbf{Z}/q\mathbf{Z})\to K_n(\mathbf{F}_{q^\prime})$ is injective.)

There are also interesting computations for mod $\ell$ coefficients. In fact, Browder showed that $K_\ast(\mathbf{Z}/q\mathbf{Z};\mathbf{Z}/\ell\mathbf{Z})$ is isomorphic as a graded ring to $(\mathbf{Z}/\ell\mathbf{Z})[\beta,\zeta]/(\zeta^2)$, where $\zeta$ and $\beta$ are generators for $K_1(\mathbf{Z}/q\mathbf{Z};\mathbf{Z}/\ell\mathbf{Z})$ and $K_2(\mathbf{Z}/q\mathbf{Z};\mathbf{Z}/\ell\mathbf{Z})$ respectively ($\beta$ is called the Bott element). If $q=p^n$ and $\ell|(q-1)$, then $\overline{\mathbf{F}_p}$ is the union of the $\mathbf{Z}/q\mathbf{Z}$, and so by the second part of the computations of the ordinary K-groups (which also holds mod $\ell$), we see that $K_\ast(\overline{\mathbf{F}_p};\mathbf{Z}/\ell\mathbf{Z})$ is the direct limit of the $K_\ast(\mathbf{Z}/p\mathbf{Z};\mathbf{Z}/\ell\mathbf{Z})$, and it turns out that $K_\ast(\overline{\mathbf{F}_p};\mathbf{Z}/\ell\mathbf{Z})$ is isomorphic to $(\mathbf{Z}/\ell\mathbf{Z})[\beta]$.

I apologize for the short post after the lengthy silence, but I will try to post more regularly soon, now that “senior” year is slowing down.

Have fun,
SKD