Remarks on p-primary torsion of the Brauer group

2024/10/13 by Yang, Yuan · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.09969

Abstract

For a smooth and proper variety X over an algebraically closed field k of characteristic p>0, the group Br(X)[p^∞] is a direct sum of finitely many copies of ℚ_p/ℤ_p and an abelian group of finite exponent. The latter is an extension of a finite group J by the group of k-points of a connected commutative unipotent algebraic group U. In this paper we show that (1) if X is ordinary, then U = 0; (2) if X is a surface, then J is the Pontryagin dual of NS(X)[p^∞]; (3) if X is an abelian variety, then J = 0. Using Crew's formula, we compute the dimension of U for surfaces and abelian 3-folds. We show that, if X is ordinary, then the unipotent subgroup of Br(X× Y) is isomorphic to the unipotent subgroup of Br(Y). Generalizing a result of Ogus, we give a criterion for the injectivity of the canonical map from flat to crystalline cohomology in degree 2.

Cited by

• Brauer groups of abelian varieties over fields of finite characteristic
• Some applications of the Nygaard filtration and quasisyntomic descent in positive characteristic

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