The inverse limit topology and profinite descent on Picard groups in K(n)-local homotopy theory

Abstract

In this paper, we study profinite descent theory for Picard groups in $K\left(n\right)$-local homotopy theory through their inverse limit topology. Building upon Burklund's result on the multiplicative structures of generalized Moore spectra, we prove that the module category over a $K\left(n\right)$-local commutative ring spectrum is equivalent to the limit of its base changes by a tower of generalized Moore spectra of type n. As a result, the $K\left(n\right)$-local Picard groups are endowed with a natural inverse limit topology. This topology allows us to identify the entire $E_1$ and $E_2$-pages of a descent spectral sequence for Picard spaces of $K\left(n\right)$-local profinite Galois extensions.

Our main examples are $K\left(n\right)$-local Picard groups of homotopy fixed points $E_n^{hG}$ of the Morava E-theory $E_n$ for all closed subgroups G of the Morava stabilizer group $G_n$. The $G=G_n$ case has been studied by Heard and Mor. At height 1, we compute Picard groups of $E_1^{hG}$ for all closed subgroups G of $G_1=Z_p^{\times }$ at all primes as a Mackey functor.