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 $K(n)$-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.

46 pages. Improved expositions and fixed typos following the referee report. Comments welcome!

References in corpus (3)

• Chromatic Cyclotomic Extensions
• Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality
• The ${\mathrm{Sp}}_{k,n}$-local stable homotopy category