Invertible objects in Franke's comodule categories

We study the Picard group of Franke's category of quasi-periodic \(E_0E\)-comodules for \(E\) a 2-periodic Landweber exact cohomology theory of height \(n\) such as Morava \(E\)-theory, showing that for \(2p-2 > n^2+n\), this group is infinite cyclic, generated by the suspension of the unit. This is analogous to, but independent of, the corresponding calculations by Hovey and Sadofsky in the \(E\)-local stable homotopy category. We also give a computation of the Picard group of \(I_n\)-complete quasi-periodic \(E_0E\)-comodules when \(E\) is Morava \(E\)-theory, as studied by Barthel--Schlank--Stapleton for \(2p-2 \ge n^2\) and \(p-1 \nmid n\), and compare this to the Picard group of the \(K(n)\)-local stable homotopy category, showing that they agree up to extension.