On Bloch’s map for torsion cycles over non-closed fields

Abstract We generalize Bloch’s map on torsion cycles from algebraically closed fields to arbitrary fields. While over is injective for zero-cycles and of codimension at most two, we show that the generalization only two but, in general, not zero-cycles. Our result implies Jannsen’s cycle class integral $\ell $ -adic continuous étale cohomology is, finitely generated This answers a question Scavia Suzuki.