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arXiv · 2406.19481

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

David Chan·Chase Vogeli
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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.

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David Chan, Chase Vogeli. 2024-06-27. The Galois-equivariant $K$-theory of finite fields. https://arxiv.org/abs/2406.19481

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