Decompositions of derived categories of gerbes and of families of Brauer-Severi varieties

It is well known that the category of quasi-coherent sheaves on a gerbe banded by a diagonalizable group decomposes according to the characters of the group. We establish the corresponding decomposition of the unbounded derived category of complexes of sheaves with quasi-coherent cohomology. This generalizes earlier work by Lieblich for gerbes over schemes whereas our gerbes may live over arbitrary algebraic stacks. By combining this decomposition with the semi-orthogonal decomposition for a projectivized vector bundle, we deduce a semi-orthogonal decomposition of the derived category of a familiy of Brauer-Severi varieties whose components can be described in terms of twisted sheaves on the base. This reproves and generalizes a result of Bernardara.