Federico Binda - GAGA type conjecture for the Brauer group via derived geometry

In Brauer III, Grothendieck considered the problem of comparing the cohomological Brauer group of a scheme X, proper and flat over a henselian DVR R, and the inverse limit of the Brauer groups lim Br(X_n), where X_n = X ⊗R R/m^n. He proved that
the canonical map Br(X) → lim Br(X_n) is injective under a number of restrictions, and left as an open problem the question on whether the formal injectivity holds in a fairly general setting. Thanks to the machinery of derived algebraic geometry and the results of Toen on derived Azumaya algebras and derived Morita theory, we are able to rephrase Grothendieck's
question in terms of a formal GAGA-type problem for smooth and proper categories, enriched over the ∞-category QCoh(X) of quasi-coherent O_X-modules. In this framework we can show that Grothendieck's injectivity conjecture always holds for a proper derived scheme X → S where S is the spectrum of any complete Noetherian local ring, if we are willing to replace the inverse limit lim Br(X_n) with the Brauer group Br(XX) of the formal scheme XX given by the colimit of the thickenings X_n. The obstruction involving the inverse system Pic(X_n) considered by Grothendieck appears naturally in the Milnor sequence for a certain tower of spaces. This
is a joint work in progress with Mauro Porta (IRMA, Strasbourg).