English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Using the pro\'etale site, we construct models for the continuous actions ofthe 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 ofas 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 forthe 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 onerecently defined by Heard; our formalism allows us to compare manydifferentials with those in the $K(n)$-local $E_n$-Adams spectral sequence, andisolate the exotic Picard elements in the $0$-stem. In particular, we show howthis recovers the computation due to Hopkins, Mahowald and Sadofsky of thegroup $\mathrm{Pic}_1$ at all primes. We also use these methods to bound theBrauer group of $K(n)$-local spectra, and compute this bound at height one.

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