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

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