The Picard group in equivariant homotopy theory via stable module categories

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.

arXiv comment37 pages

References in corpus (5)

• Nilpotence and descent in equivariant stable homotopy theory
• The Picard group of topological modular forms via descent theory
• Endotrivial modules for finite groups via homotopy theory
• On the parametrized Tate construction and two theories of real $p$-cyclotomic spectra
• Torus actions on stable module categories, Picard groups, and localizing subcategories

Cited by in corpus (4)

• Derived Mackey functors and $C_{p^n}$-equivariant cohomology
• Dualizable objects in stratified categories and the 1-dimensional bordism hypothesis for recollements
• Stratifying integral representations of finite groups
• The slice spectral sequence for the cyclic group of order p