  In this paper, we generalize a result of Karpenko on the torsion in the
second quotient of the gamma filtration for Severi-Brauer varieties to higher
degrees. As an application, we provide a nontrivial torsion in higher Chow
groups and the topological filtration of the associated generic variety and
obtain new upper bounds for the annihilators of the torsion subgroups in the
Chow groups of a large class of Severi-Brauer varieties. In particular, using
the torsion in higher degrees, we show indecomposability of certain algebras.
