It was shown in "Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated.", that the Chow group mod algebraic equivalence of smooth complex varieties are not necessarily finitely generated. My question is whether it is possible for some smooth quasi-projective complex variety, its Chow group mod the algebraic equivalence contains a non-torsion infinitely divisible element? (more precisely a non-torsion element that can be divided by some prime $l$ infinite number of times.)
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