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We study the injectivity of the cycle class map with values in Jannsen'scontinuous \'etale cohomology, by using refinements that go through \'etalemotivic cohomology and the ``tame'' version of Jannsen's cohomology. Inparticular, we use this to show that the Tate and the Beilinson conjecturesimply that its kernel is torsion in positive characteristic, and to revisitrecent counterexamples to injectivity.

On the injectivity and non-injectivity of the $l$-adic cycle class maps