The Picard group of the category of $C_n$-equivariant stable homotopy theory

Abstract

For a finite group $G$, there is a map $RO(G)\to \mathrm{Pic}(S{p}^G)$ from the real representation ring of $G$ to the Picard group of $G$-spectra. This map is not known to be surjective in general, but we prove that when $G$ is cyclic this map is indeed surjective and in that case we describe $\mathrm{Pic}(S{p}^G)$ explicitly. We also show that for an arbitrary finite group $G$ homology and cohomology with coefficients in a cohomological Mackey functor do not see the part of $\mathrm{Pic}(S{p}^G)$ coming from the Picard group of the Burnside ring. Hence these homology and cohomology calculations can be graded on a smaller group.

arXiv comment13 pages. Comments welcome