Some 35 years ago, Ken Ribet proved that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. In joint work with Damian Roessler, we show that Ribet's theorem is an instance of a general cohomological statement about smooth projective varieties over K. We also present a largely conjectural generalization to torsion cycles of higher codimension, as well as an analogue in positive characteristic. Speaker(s): Tamas Szamuely (Alfred Renyi Institute of Mathematics)