Picard Groups in Equivariant Algebra and Stable Homotopy Theory

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.