Invertible objects in Franke's comodule categories

Abstract

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.