  We show that the torsion in the group of indecomposable $(2,1)$-cycles on a
smooth projective variety over an algebraically closed field is isomorphic to a
twist of its Brauer group, away from the characteristic. In particular, this
group is infinite as soon as $b_2-\rho>0$. We derive a new insight into
Roitman's theorem on torsion $0$-cycles over a surface.
