We compute the Picard group of the category of –local module spectra over the ring spectrum , where is a height Morava –theory and is a subgroup of the associated Morava stabilizer group. This group can be identified with the Picard group of –local –modules in genuine –spectra. We show that in addition to a cyclic subgroup of order generated by , the Picard group contains a subgroup of order generated by , where is the sign representation of the group . In the process, we completely compute the –graded Mackey functor homotopy fixed point spectral sequence for the –spectrum .
"Invertible $K(2)$–local $E$–modules in $C_4$–spectra." Algebr. Geom. Topol. 20 (7) 3423 - 3503, 2020. https://doi.org/10.2140/agt.2020.20.3423