Modular fixed points in equivariant homotopy theory

TL;DR

This paper establishes an equivalence between derived categories of permutation modules and modules over certain spectra in equivariant homotopy theory, introducing a modular fixed point functor and analyzing the Picard group for p-groups.

Contribution

It defines a modular fixed point functor in equivariant spectra and identifies it with a known functor, linking homotopy theory and representation theory in a novel way.

Findings

Equivalence between derived permutation modules and modules over Eilenberg-MacLane spectra.

Identification of the modular fixed point functor with the one introduced by Balmer-Gallauer.

Description of the Picard group for p-groups via class functions satisfying Borel-Smith conditions.

Abstract

We show that the derived $\infty$-category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the $\infty$-category of equivariant spectra. On such module categories we define a modular fixed point functor using geometric fixed points followed by an extension of scalars and identify it with the modular fixed point functor on derived permutation modules introduced by Balmer-Gallauer. As an application, we show that the Picard group of such a module category for a $p$-group is given by the group of class functions satisfying the Borel-Smith conditions. In the language of representation theory, this result was first obtained by Miller.