Invertible $K(2)$–local $E$–modules in $C_4$–spectra

Abstract

We compute the Picard group of the category of $K\left(2\right)$–local module spectra over the ring spectrum $E^{h{C}_4}$, where $E$ is a height $2$ Morava $E$–theory and $C_4$ is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of $K\left(2\right)$–local $E$–modules in genuine $C_4$–spectra. We show that in addition to a cyclic subgroup of order $32$ generated by $E\wedge S^1$, the Picard group contains a subgroup of order $2$ generated by $E\wedge S^{7+\sigma }$, where $\sigma$ is the sign representation of the group $C_4$. In the process, we completely compute the $RO\left(C_4\right)$–graded Mackey functor homotopy fixed point spectral sequence for the $C_4$–spectrum $E“$.