Separable commutative algebras in equivariant homotopy theory

Abstract

Given a finite group $G$ and a commutative ring $G$-spectrum $R$, we study the separable commutative algebras in the category of compact $R$-modules. We isolate three conditions on the geometric fixed points of $R$ which ensure that every separable commutative algebra is standard, i.e. arises from a finite $G$-set. In particular we show that all separable commutative algebras in the categories of compact objects in $G$-spectra and in derived $G$-Mackey functors are standard provided that $G$ is a $p$-group. In these categories we also show that for a general finite group $G$, not all separable commutative algebras are standard. We finally discuss how the classification of separable commutative algebras in compact $G$-spectra varies if we require the existence of multiplicative norms. We show that if $G$ is solvable, then any separable commutative algebra therein that is normed is automatically standard. However, if $G$ is not solvable, we provide examples of separable commutative algebras that are normed but not standard.