#### Volume 20, issue 7 (2020)

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

### Agnès Beaudry, Irina Bobkova, Michael Hill and Vesna Stojanoska

Algebraic & Geometric Topology 20 (2020) 3423–3503
##### 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 $E“$.

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##### Keywords
chromatic homotopy theory, Morava E–theory, Picard groups, higher real K–theory
##### Mathematical Subject Classification 2010
Primary: 55P42, 55Q91
Secondary: 20J06, 55M05, 55P60, 55Q51