GAGA type conjecture for the Brauer group via derived geometry

by Federico Binda (University of Milan)

Thursday 25 Nov 2021, 13:30 → 15:30 Europe/Moscow

818-1526-4739 (Zoom)

In Brauer III, Grothendieck considered the problem of comparing the cohomological Brauer group $Br(X) = H^2_{et}(X, \mathbb{G}_m)$ of a scheme $X$, proper and flat over a henselian DVR $R$, and the inverse limit of the Brauer groups $\lim_n Br(X_n)$ where  $X_n=X\otimes R/ \mathfrak{m}^n$. He proved that the canonical map $Br(X) \to \lim_n Br(X_n)$ 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 $\infty$-category $QCoh(X)$ of quasi-coherent $\mathcal{O}_X$ modules. In this framework we can show that Grothendieck's injectivity conjecture always holds for a proper derived scheme $X\to S$, where $S$ is the spectrum of any complete Noetherian local ring,  if we are willing to replace the inverse limit $\lim_n Br(X_n)$ with the Brauer group $Br(\mathfrak{X})$ of the formal scheme $\mathfrak{X}$ given by the colimit of the thickenings $X_n$.  We also show that the original question of Grothendieck has in general a negative answer, by constructing an explicit counterexample. This is a joint work with Mauro Porta (IRMA, Strasbourg).

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