Algebraic models of change of groups functors in (co)free rational equivariant spectra (Q2153811)

An algebraic model for a stable model category \(\mathcal{M}\) is a model category \(\mathcal{C}\) built from simple algebraic terms such that \(\mathcal{M}\) is Quillen equivalent to \(\mathcal{C}\). The purpose is to make calculations and constructions easier by working in the algebraic model. In the case of rational \(G\)-spectra, algebraic models have been constructed for many different groups, including all finite groups, \(SO(2)\), \(O(2)\) and \(SO(3)\). By restricting to certain special classes of spectra there are constructions of algebraic models for whole classes of groups. This paper focuses on the case of (co)free rational \(G\)-spectra, which are the representing objects for cohomology theories on free equivariant spaces. An algebraic model is known for every compact Lie group \(G\) in the free case by work of \textit{J. P. C. Greenlees} and \textit{B. Shipley} [Bull. Lond. Math. Soc. 46, No. 1, 133--142 (2014; Zbl 1294.55002)]. The cofree case is covered by \textit{L. Pol} and \textit{J. Williamson} [J. Pure Appl. Algebra 224, No. 11, Article ID 106408, 32 p. (2020; Zbl 1446.55009)]. These algebraic models (at least for connected \(G\)) are given in terms of complete and torsion modules over the graded polynomial ring \(H^*(BG)\). For \(H\) a subgroup of \(G\) (both connected), the current paper constructs (four) adjoint functors between the algebraic models for \(G\) and \(H\). The main result of the paper is a comparison between these new functors and the change-of-groups adjunction (forgetful, induced, coinduced) between free rational \(G\)-spectra and free rational \(H\)-spectra. A stronger result when \(H\) and \(G\) have the same rank is given. The main result depends on new technology comparing Quillen functor pairs across zig-zags of Quillen equivalences (based on the mates correspondence). The good monoidal properties of the zig-zag relating free rational \(G\)-spectra to the algebraic model play a major role in the result, as does the Wirthmüller isomorphism.