Brauer groups of 1-motives

Abstract

Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group H e ´ t 2 ( M , G m , M ) of a 1-motive M , whose pull-back via the unit section ϵ : S → M is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of H e ´ t 2 ( M , G m , M ) come from Azumaya algebras .