Picard and Brauer groups of $K(n)$-local spectra via profinite Galois descent

TL;DR

This paper develops a profinite Galois descent framework for $K(n)$-local spectra, enabling detailed analysis of Picard and Brauer groups, and relating spectral sequences to known Adams spectral sequences.

Contribution

It introduces a new formalism using the proétale site to study Galois actions on Morava E-theory and Picard spectra, connecting descent spectral sequences with Adams spectral sequences.

Findings

Computed the Picard group $ ext{Pic}_1$ at all primes.

Bounded the Brauer group of $K(n)$-local spectra at height one.

Related descent spectral sequences to known Adams spectral sequences.

Abstract

Using the pro\'etale site, we construct models for the continuous actions of the Morava stabiliser group on Morava E-theory, its $\infty$-category of $K(n)$-local modules, and its Picard spectrum. For the two sheaves of spectra, we evaluate the resulting descent spectral sequences: these can be thought of as homotopy fixed point spectral sequences for the profinite Galois extension $L_{K(n)} \mathbb S \to E_n$. We show that the descent spectral sequence for the Morava E-theory sheaf is the $K(n)$-local $E_n$-Adams spectral sequence. The spectral sequence for the sheaf of Picard spectra is closely related to one recently defined by Heard; our formalism allows us to compare many differentials with those in the $K(n)$-local $E_n$-Adams spectral sequence, and isolate the exotic Picard elements in the $0$-stem. In particular, we show how this recovers the computation due to Hopkins, Mahowald and Sadofsky of the group $\mathrm{Pic}_1$ at all primes. We also use these methods to bound the Brauer group of $K(n)$-local spectra, and compute this bound at height one.