GAGA type conjecture for the Brauer group via derived geometry

Abstract: In Brauer III, Grothendieck considered the problem of comparing the cohomological Brauer group $Br(X) = H^2_{et}(X,G_m)$ of a scheme $X$, proper and flat over a henselian DVR $R$, and the inverse limit of the Brauer groups $\lim_nBr(X_n)$, where $X_n = X\otimes_R R/m^n$. 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 To\"en 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 $O_X$-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 $Br(X)$ of the formal scheme $\mathfrak{X}$ 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).

commutative algebraalgebraic geometryalgebraic topologygeometric topologyK-theory and homology

Audience: researchers in the topic

electronic Algebraic K-theory Seminar

Series comments: Description: Research seminar on algebraic K-theory