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

 

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