Affine étale group schemes over Tambara fields

Abstract

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 $\underline{K}\to \mathrm{FP}(L)$ 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 $G=C_p$.

18 pages, comments welcome! V2 comments: slightly strengthened some results