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If k is separably closed, finite or p-adic, this describes it as an extension of a finite group F by a divisible group D, where F is the torsion subgroup of the cokernel of the l-adic cycle map. If k is finite and X is projective and of abelian type, verifying the Tate conjecture, D=0. If k is separably closed, we relate D to an l-adic Griffiths group. 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