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Brauer groups of 1-motives. (English) Zbl 1475.14042

Summary: Over a normal base scheme, we prove the generalized Theorem of the Cube for 1-motives and that a torsion class of the group \(\mathrm{H}_{\text{èt}}^2(M,\mathbb{G}_{m,M})\) of a 1-motive \(M\), whose pull-back via the unit section \(\epsilon : S \to M\) is zero, comes from an Azumaya algebra. In particular, we deduce that over an algebraically closed field of characteristic zero, all classes of \(\mathrm{H}_{\text{èt}}^2(M, \mathbb{G}_{m,M})\) come from Azumaya algebras.

MSC:

14F22 Brauer groups of schemes
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16K50 Brauer groups (algebraic aspects)
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