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 \'etale cohomology with $\bf Q/\bf
Z$-coefficients of a smooth proper $\overline K$-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.