Modular fixed points in equivariant homotopy theory
arXiv:2506.21413
The paper shows that the derived ∞‑category of permutation modules is equivalent to modules over the Eilenberg‑MacLane spectrum of a constant Mackey functor in equivariant spectra, defines a modular fixed point functor via geometric fixed points, and uses it to compute Picard groups for p‑groups in terms of class functions satisfying Borel‑Smith conditions.
Abstract
We show that the derived -category of permutation modules is equivalent to the category of modules over the Eilenberg-MacLane spectrum associated to a constant Mackey functor in the -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 -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.
52 pages; v2: minor changes, accepted version