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.