On the equivariant $KU_G$-local sphere for finite abelian groups

Abstract

Given a finite abelian group $G$ and a Sylow $p$-subgroup $N_p$, we prove that the $KU_G/p$-local sphere spectrum is equivalent to the homotopy fixed points of a $p$-complete $KO_{N_p}$-module spectrum. Then we compute the $\mathbb{Z}$-graded homotopy Mackey functors of the $KU_G$-local sphere spectrum. This result generalizes the computation of arXiv:2303.12271 for finite $p$-groups, where $p$ is an odd prime. Finally, by comparing the Bousfield classes of $KU_G/p$ and $G$-equivariant Morava $K$-theory, we prove that the $KU_G/p$-local sphere spectrum is equivalent to a wedge sum of equivariant Morava $K$-theory localized sphere spectra, and describe the $RO(G)$-graded homotopy Mackey functors of the $KU_G/p$-local sphere spectrum.