Separable commutative algebras in equivariant homotopy theory

TL;DR

This paper investigates the structure of separable commutative algebras in equivariant homotopy theory, identifying conditions under which they are standard and exploring how group properties influence their classification.

Contribution

It introduces conditions on geometric fixed points that determine when separable commutative algebras are standard, especially for $p$-groups and solvable groups, and examines the impact of multiplicative norms.

Findings

All separable commutative algebras are standard for $p$-groups.

Not all separable commutative algebras are standard for general finite groups.

Normed separable commutative algebras are standard if $G$ is solvable, but not necessarily if $G$ is non-solvable.

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.