Classifying rational $G$-spectra for finite $G$

Abstract

We give a new proof that for a finite group G, the category of rational Gequivariant spectra is Quillen equivalent to the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Furthermore the Quillen equivalences of our proof are all symmetric monoidal. Thus we can understand categories of algebras or modules over a ring spectrum in terms of the algebraic model. A G-equivariant cohomology theory E ∗ is said to be rational if E ∗ (X) is a rational vector space for every G-space X. For G, a finite group, we want to describe the category of rational G-equivariant cohomology theories in terms of a simple algebraic model. In particular, we want to understand those cohomology theories with a multiplication and the modules over such a theory. To do so, we give a particular construction of a model category of G-spectra whose homotopy category is (equivalent to) the category of rational G-equivariant cohomology theories. We show that this model category is symmetric monoidally Quillen equivalent to an explicit algebraic model: the product of the model categories of chain complexes of modules over the rational group ring of the Weyl group of H in G, as H runs over the conjugacy classes of subgroups of G. Since our Quillen equivalences are symmetric monoidal, the category of ring spectra is Quillen equivalent to the category of monoids in the algebraic model. Let WGH be the Weyl group of H in G, the quotient of the normaliser of H in G by H, then a monoid in the category of chain complexes of modules over the rational group ring of WGH, is a differential graded Q-algebra with an action (through algebra maps) of the group WGH. The category of modules over a ring spectrum will then be Quillen equivalent to modules over a monoid in the algebraic model. So, if one has a ring spectrum R that one wishes to study, one can look at its image ˜ R