The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory
Guchuan Li, Ningchuan Zhang
TL;DR
This work develops a pro-structural framework for Picard groups in $K(n)$-local homotopy theory by lifting Picard groups to inverse limits along towers of generalized Moore spectra of type $n$, yielding a natural inverse limit topology on $\mathrm{Pic}_{K(n)}(R)$. It proves a Grothendieck-existence-type theorem: for a $K(n)$-local $\mathbb{E}_\infty$-ring $R$, the module category $\mathsf{Mod}_{K(n)}(R)$ is equivalent to the limit of base changes $\lim_j\mathsf{Mod}(R\wedge M_j)$, and consequently $\mathfrak{pic}_{K(n)}(R)\simeq\lim_j\mathfrak{pic}(R\wedge M_j)$ with a topology independent of the tower. The paper introduces profinite descent spectral sequences for $K(n)$-local Picard spaces associated to descendable profinite Galois extensions, identifying the $E_1$ and $E_2$ pages and showing the differentials align with the continuous group cohomology cobar complex. Applications focus on Picard groups of Morava $E$-theory fixed points $E_n^{hG}$, including algebraicity results for exotic Picard groups $\kappa(E_n^{hG})$ and descent filtrations that recover known height-one phenomena in the appropriate cases. At height one, detailed Mackey functor computations yield explicit $\mathrm{Pic}_{K(1)}(E_1^{hG})$ for all closed subgroups $G\le\mathbb{Z}_p^{\times}$, with a contrast between odd primes (topological generators) and $p=2$ (more intricate, non-cyclic behavior, including the exotic element).
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.