Picard Groups in Equivariant Algebra and Stable Homotopy Theory

TL;DR

This paper explores the structure of Picard groups in equivariant stable homotopy theory, establishing isomorphisms and classifications of invertible Mackey functors and modules for finite abelian groups.

Contribution

It proves the folklore isomorphism between Picard groups of Burnside rings and Mackey functors for finite groups, and classifies invertible Mackey functors and modules for finite abelian groups.

Findings

Established the folklore isomorphism for finite groups.

Classified invertible Mackey functors for finite abelian groups.

Classified invertible modules over the Burnside ring for finite abelian groups.

Abstract

Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the equivariant stable homotopy category. Though there is a canonical map of abelian groups $\text{RO}(G) \rightarrow \text{Pic}(\text{Ho}(\text{Sp}^G))$ relating the two, this map is neither injective or surjective in general. Fausk, Lewis, and May give an algebraic expression of $\text{Pic}(\text{Ho}(\text{Sp}^G))$ in terms of the Picard group of the Burnside ring $A(G)$, and this work suggests a folklore isomorphism between $\text{Pic}(A(G))$ and $\text{Pic}(\text{Mack}_G)$. We prove the existence of this folklore isomorphism in the setting of finite groups, then leverage our analysis to prove a classification of invertible Mackey functors in the setting of finite abelian groups. As a consequence, we furnish a classification of invertible $A(G)$-modules again for $G$ a finite abelian group.