Affine \'{e}tale group schemes over Tambara fields

TL;DR

This paper classifies finite étale extensions and affine étale group schemes over Tambara functors, establishing Galois descent and discovering new families of étale extensions, thus advancing the understanding of Tambara functor algebraic structures.

Contribution

It provides a complete classification of étale extensions over G-Tambara functors and introduces new families of étale extensions, extending previous results to more general group actions.

Findings

Classified all finite étale extensions over G-Tambara functors.

Established Galois descent for Tambara functor algebraic closures.

Identified new families of étale extensions and generalized existing results.

Abstract

We classify finite \'{e}tale extensions and finite affine \'{e}tale 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 \'{e}tale extensions of any $G$-Tambara functor and show that, together with one of the families discovered by Lindenstrauss--Richter--Zou, these give all finite \'{e}tale 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 \'{e}tale, generalizing a result of Lindenstrauss--Richter--Zou when $G$ is cyclic. Lastly, we classify flat finitely generated $\underline{\mathbb{F}}$-modules when $G = C_p$.