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

TL;DR

This paper explores the inverse limit topology on Picard groups in $K(n)$-local homotopy theory, establishing a descent spectral sequence and computing specific Picard groups at height 1 for all primes.

Contribution

It introduces a novel inverse limit topology on $K(n)$-local Picard groups and applies profinite descent theory to compute these groups in new cases.

Findings

Identified the inverse limit topology on $K(n)$-local Picard groups.

Established a descent spectral sequence for Picard spaces in this setting.

Computed Picard groups of $E_1^{hG}$ at height 1 for all primes.

Abstract

In this paper, we study profinite descent theory for Picard groups in $K(n)$-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 $K(n)$-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 $K(n)$-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 $K(n)$-local profinite Galois extensions. Our main examples are $K(n)$-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 $\mathbb{G}_n$. The $G=\mathbb{G}_n$ 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 $\mathbb{G}_1=\mathbb{Z}_p^\times$ at all primes as a Mackey functor.