ホッジ予想
have shown that in order to obtain a correct integral Hodge conjecture, one needs to replace Chow groups, which can also be expressed as motivic cohomology groups, by a variant known as "étale" (or "Lichtenbaum") "motivic cohomology". They show that the rational Hodge conjecture is equivalent to an integral Hodge conjecture for this modified motivic cohomology.