  We study the injectivity of the cycle class map with values in Jannsen's
continuous \'etale cohomology, by using refinements that go through \'etale
motivic cohomology and the ``tame'' version of Jannsen's cohomology. In
particular, we use this to show that the Tate and the Beilinson conjectures
imply that its kernel is torsion in positive characteristic, and to revisit
recent counterexamples to injectivity.
