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.