Approximations of Eilenberg-MacLane spaces by smooth projective varieties

Abstract

In this note we construct approximations by smooth projective varieties of some Eienberg-MacLane spaces in the $A^1$-homotopy category. Using these, we study the cycle maps from Chow rings to etale cohomology rings.

Lemma 4.1 in the section 4 was error, So we can not show the main lemma Lemma 4.2. Moreover we can not prove main theorems Theorem 1.1-1.4 in the introduction