Invertible objects in Franke's comodule categories (Q6561304)

Chromatic homotopy theory studies localizations of the \(p\)-local stable homotopy category with respect to the Morava \(K(n)\)-theories and the Johnson-Wilson spectra \(E(n)\). These categories are known to be more algebraic in nature than the whole stable homotopy category. Unpublished work of \textit{J. Franke} [``Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence'', Unpublished Preprint, available at \url{https://faculty.math.illinois.edu/K-theory/0139/} (1996)] constructs an algebraic model for the \(E(n)\)-local stable homotopy category, which is an equivalence when \(n=1\) and \(p>2\).\N\NFollowing Franke's construction this paper studies quasi-periodic derived categories of comodules over a Hopf algebroid. These algebraic categories are symmetric monoidal and the main results of this paper are on their Picard groups -- the group of objects which are units under the monoidal product.\N\NThe author defines the notion of a descendable Hopf algebroid and uses this to construct an equivalence of stable infinity categories between the quasi-perodic comodule category and a totalisation of a complex of categories of modules. This form of descent is compatible with taking Picard spectra and hence gives a calculation of the Picard group (which is \(\pi_0\) of the Picard spectrum) of the quasi-periodic comodule category.\N\NMore specifically, when \(2p - 2 > n^2 + n\), the Picard group of quasi-periodic \(E_0 E\)-comodules for \(E\) a 2-periodic Landweber exact cohomology theory of height \(n\) (such as Morava \(E\)-theory) is infinite cyclic, generated by the suspension of the unit. Some results along this line have been proven by \textit{D. Barnes} and \textit{C. Roitzheim} [Adv. Math. 228, No. 6, 3223--3248 (2011; Zbl 1246.55009)] and \textit{T. Barthel} et al. [Invent. Math. 220, No. 3, 737--845 (2020; Zbl 1442.55002)]. However, those results rely on calculations from \(E(n)\)-local stable homotopy theory.\N\NFollowing Barthel, Schlank and Stapleton [loc. cit.], the author further studies the \(K(n)\)-local equivalent using the \(I_n\)-completion of quasi-periodic comodule category. In the range \(2p - 2 \ge n^2\) with \(p - 1 \mid n\) the author shows that the Picard group of the \(K(n)\)-local stable homotopy category agrees with that of \(I_n\)-complete quasi-periodic \(E_0E\)-comodules, up to extension. This is analogous to, but independent of, the corresponding calculations by \textit{M. Hovey} and \textit{H. Sadofsky} [J. Lond. Math. Soc., II. Ser. 60, No. 1, 284--302 (1999; Zbl 0947.55013)].