The Balmer spectrum of rational equivariant cohomology theories

Abstract

The category of rational G-equivariant cohomology theories for a compact Lie group $G$ is the homotopy category of rational G-spectra and therefore tensor-triangulated. We show that its Balmer spectrum is the set of conjugacy classes of closed subgroups of $G$, with the topology corresponding to the topological poset introduced in the author's study of rational Mackey functors. This is used to classify the collections of subgroups arising as the geometric isotropy of finite $G$-spectra. The ingredients for this classification are (i) the algebraic model of free spectra of the author and B.Shipley (arXiv 1101.4818), (ii) the Localization Theorem of Borel-Hsiang-Quillen and (iii) tom Dieck's calculation of the rational Burnside ring.