Cohomology and torsion cycles over the maximal cyclotomic extension

A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the ?tale cohomology with Q/Z-coefficients of a smooth proper K-variety defined over K. We also present a conjectural generalization of Ribet?s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.