The Galois-equivariant $K$-theory of finite fields

TL;DR

This paper computes the equivariant algebraic K-theory of finite fields with Galois group actions, revealing a splitting structure and explicit descriptions, especially for prime order Galois groups.

Contribution

It provides the first explicit computation of the RO(G)-graded equivariant K-theory of finite fields with Galois actions, including a splitting result and prime order case descriptions.

Findings

K-groups split into computable and known parts

Equivariant K-theory shares Tate and fixed point spectra with Eilenberg--MacLane spectrum

Explicit presentation for prime order Galois groups

Abstract

We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied $RO(G)$-graded coefficient groups of the equivariant Eilenberg--MacLane spectrum $H\underline{\mathbb Z}$. Our comparison between the equivariant $K$-theory spectrum and $H\underline{\mathbb Z}$ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where $G$ has prime order, we provide an explicit presentation of the equivariant $K$-groups.