The Picard group in equivariant homotopy theory via stable module categories

TL;DR

This paper introduces a new method for analyzing invertible G-spectra using stable module categories, providing a complete classification for G=A_5 and relating Picard groups across different categories.

Contribution

It develops an isotropy separation technique for compact objects and applies it to classify invertible G-spectra, also linking Picard groups of different categories.

Findings

Complete analysis of invertible G-spectra for G=A_5

Equivalence of Picard groups between Sp^G and derived Mackey functors

Explicit description of invertible G-spectra via geometric fixed points

Abstract

We develop a mechanism of "isotropy separation for compact objects" that explicitly describes an invertible $G$-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module category. As an application, we carry out a complete analysis of invertible G-spectra in the case $G=A_5$. A further application is given by showing that the Picard groups of $\mathrm{Sp}^G$ and a category of derived Mackey functors agree.