Algebraic models and change of groups for equivariant spectra

In this thesis we consider the interaction between extra structures present in category of rational free G-spectra, namely the monoidal structure and the change of groups adjunctions, and the passage to the algebraic model. We prove that the category of rational cofree G-spectra admits a monoidal algebraic model in terms of L-complete modules. In order to prove this, we develop the Left Localization Principle which gives mild conditions under which a Quillen adjunction descends to a Quillen equivalence between left Bousfield localizations.

We give a model categorical argument showing that the induction, restriction and coinduction functors between categories of (co)free rational equivariant spectra correspond to functors between the algebraic models for connected compact Lie groups. In order to do this, we provide general tools to check whether Quillen functors correspond. Since the construction of algebraic models relies upon the fact that polynomial rings are strongly intrinsically formal as commutative DGAs, we pay careful attention to the interaction of model structures on commutative algebras and modules. In particular, we prove that Shipley’s algebraicization theorem respects the extra compatibility between commutative algebras and modules in the flat model structure on spectra.