The Galois-equivariant $K$-theory of finite fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913573030068224 |
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| author | Chan, David Vogeli, Chase |
| author_facet | Chan, David Vogeli, Chase |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_19481 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Galois-equivariant $K$-theory of finite fields Chan, David Vogeli, Chase K-Theory and Homology Algebraic Topology 19D50 (Primary), 19L47, 55P91, 55Q91 (Secondary) 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. |
| title | The Galois-equivariant $K$-theory of finite fields |
| topic | K-Theory and Homology Algebraic Topology 19D50 (Primary), 19L47, 55P91, 55Q91 (Secondary) |
| url | https://arxiv.org/abs/2406.19481 |