Invertible objects in Franke's comodule categories

TL;DR

This paper analyzes the Picard group of Franke's category of quasi-periodic E_0E-comodules for certain Landweber theories, showing it is infinite cyclic under specific conditions and comparing it to the Picard group in the stable homotopy category.

Contribution

It provides the first computation of the Picard group of Franke's comodules for Morava E-theories and relates it to known Picard groups in stable homotopy theory.

Findings

Picard group is infinite cyclic for 2p-2 > n^2+n.

Computed Picard group of I_n-complete quasi-periodic E_0E-comodules.

Picard groups of these categories agree with those of the K(n)-local stable homotopy category up to extension.

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.