Affine étale group schemes over Tambara fields
Noah Wisdom
TL;DR
This work develops the commutative algebra of Tambara functors to classify finite étale extensions and finite affine étale group schemes over the $G$-Tambara functor $\underline{\mathbb{F}}$ for algebraically closed $\mathbb{F}$ and finite $G$. It establishes $G$-Galois descent along $\underline{\mathbb{F}} \to \mathrm{CoInd}_e^G \mathbb{F}$ and shows that finite étale extensions are exactly finite products of base-change from subgroups via $\underline{\mathbb{F}} \to \mathrm{CoInd}_H^G \underline{\mathbb{F}}$, with the fixed-point functor FP providing an equivalence to étale objects over $\mathbb{F}$ with a $G$-action. The paper proves stability and base-change properties of étale morphisms, including the fundamental result that the bottom-level map detects étaleness, and proves that $\underline{K} \to \mathrm{FP}(L)$ is étale for any $G$-Galois extension $L/K$. In the modular setting with $G=C_p$, it also classifies flat finitely generated $\underline{\mathbb{F}}$-modules, showing flat implies free. Together, these results yield a complete, equivariant analogue of étale theory for Tambara functors and connect to Galois-descent phenomena in homotopy-theoretic contexts.
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 $\underline{K} \rightarrow \mathrm{FP}(L)$ 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 $G = C_p$.