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

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.