An algebraic model for rational toral $G$–spectra

Abstract

For $G$ a compact Lie group, toral $G$–spectra are those rational $G$–spectra whose geometric isotropy consists of subgroups of a maximal torus of $G$. The homotopy category of rational toral $G$–spectra is a retract of the category of all rational $G$–spectra.

We show that the abelian category of Greenlees (Algebr. Geom. Topol. 16 (2016) 1953–2019) gives an algebraic model for the toral part of rational $G$–spectra. This is a major step in establishing an algebraic model for all rational $G$–spectra for any compact Lie group $G$.