Smoothness of the formal Brauer group

Abstract: Let X be a smooth and proper variety over an algebraically closed field of characteristic p. The formal Brauer group of X is the functor which parametrizes deformations of the trivial Brauer class of X. Under mild assumptions, it is representable by a formal group, closely related to the p-torsion of Br(X). We will give criteria for this formal group to be smooth in terms of the crystalline cohomology of X, thus providing a partial answer to a question of Artin-Mazur. The strategy is to relate the formal Brauer group to crystalline cohomology using the relationship between fppf cohomology, crystalline cohomology and the Nygaard filtration. These criteria can be used in practice to produce varieties with non-smooth formal Brauer group, which are constructed as higher-dimensional analogues of Igusa's surface with non-smooth Picard group.

number theory

Audience: researchers in the topic

London number theory seminar

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