Affine étale group schemes over Tambara fields
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918171974303744 |
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| author | Wisdom, Noah |
| author_facet | Wisdom, Noah |
| contents | We classify finite étale extensions and finite affine étale group schemes over the $G$-Tambara functor $\underline{\mathbb{F}}$, for $\mathbb{F}$ any algebraically closed field and $G$ any finite group. This establishes $G$-Galois descent from the Tambara functor algebraic closure of $\underline{\mathbb{F}}$. In particular, we find new families of étale extensions of any $G$-Tambara functor and show that, together with one of the families discovered by Lindenstrauss--Richter--Zou, these give all finite étale extensions of $\underline{\mathbb{F}}$. Our arguments also show that the map $\underline{K} \rightarrow \mathrm{FP}(L)$ associated to any $G$-Galois extension $L$ of $K$ is étale, generalizing a result of Lindenstrauss--Richter--Zou when $G$ is cyclic. Lastly, we classify flat finitely generated $\underline{\mathbb{F}}$-modules when $G = C_p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09365 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Affine étale group schemes over Tambara fields Wisdom, Noah Algebraic Topology Algebraic Geometry 14B25 (primary) 55P91, 14L15 (secondary) We classify finite étale extensions and finite affine étale group schemes over the $G$-Tambara functor $\underline{\mathbb{F}}$, for $\mathbb{F}$ any algebraically closed field and $G$ any finite group. This establishes $G$-Galois descent from the Tambara functor algebraic closure of $\underline{\mathbb{F}}$. In particular, we find new families of étale extensions of any $G$-Tambara functor and show that, together with one of the families discovered by Lindenstrauss--Richter--Zou, these give all finite étale extensions of $\underline{\mathbb{F}}$. Our arguments also show that the map $\underline{K} \rightarrow \mathrm{FP}(L)$ associated to any $G$-Galois extension $L$ of $K$ is étale, generalizing a result of Lindenstrauss--Richter--Zou when $G$ is cyclic. Lastly, we classify flat finitely generated $\underline{\mathbb{F}}$-modules when $G = C_p$. |
| title | Affine étale group schemes over Tambara fields |
| topic | Algebraic Topology Algebraic Geometry 14B25 (primary) 55P91, 14L15 (secondary) |
| url | https://arxiv.org/abs/2508.09365 |