This paper introduces a class of genuine \(G\)-spectra that are highly amenable to equivariant calculations. The author shows that this class is closed under the most common (equivariant) operations and that examples of this class occur naturally and quite widely. The rest of the paper examines the various equivariant calculations that simplify on this class.
Let \(R\) be an \(E_\infty\)-monoid in genuine \(G\)-spectra. That is, the multiplication of \(R\) is associative and commutative up to all higher homotopies, but only up to the lowest level of equivariant commutativity in the sense of A. J. Blumberg and M. A. Hill [Adv. Math. 285, 658–708 (2015; Zbl 1329.55012)]. A \(G\)-spectrum \(E\) is said to be \(R\)-free if \(R \wedge E\) splits as a wedge of \(R\)-modules of the form \[ R \wedge \left( G_+ \wedge_H S^V \right) \] for \(V\) a virtual representation of \(H\). The class of \(R\)-free \(G\)-spectra is closed under coproducts, restriction under (arbitrary) change of groups, induction from a subgroup and the smash product. Moreover, if \(R\) is a \(G\)-\(E_\infty\)-ring spectrum (the highest level of equivariant commutativity), free \(G\)-spectra are closed under norm maps. Finally, with additional finiteness conditions, free spectra are closed under taking duals.
The author proves numerous results about how \(R\)-free \(G\)-spectra have excellent computational properties. For example, these spectra admit a Künneth theorem and an equivariant version of Snaith’s theorem. By adding further assumptions on the monoidal structure of \(R\) and \(E\), the author shows how the usual construction of Hopf algebroids and comodules from ring spectra can be lifted to take place in the category of \(RO(G)\)-graded Tambara functors.
In the final section, the author defines the related notions of pure and isotropic spectra as \(H \underline{\mathbb{Z}}\)-free \(G\)-spectra that decompose into slice spheres satisfying certain properties. These spectra are even better behaved computationally, as demonstrated by computations of the full Tambara and co-Tambara functor structures on the homology of \(BU_{\mathbb{R}}\) and \(\mathrm{Map}^{C_2} (G, BU_{\mathbb{R}})\). The paper finishes with applications of the theory to the Rothenberg-Steenrod and Eilenberg-Moore spectral sequences, illustrated by homology computations of \(BBU_{\mathbb{R}}\) and \(\mathrm{Map}^{C_2} (G, BBU_{\mathbb{R}})\).