Approximations of Eilenberg-MacLane spaces by smooth projective varieties

TL;DR

This paper constructs smooth projective varieties approximating certain Eilenberg-MacLane spaces within the $A^1$-homotopy framework, enabling analysis of cycle maps from Chow rings to etale cohomology rings.

Contribution

It introduces a method to approximate Eilenberg-MacLane spaces with smooth projective varieties in the $A^1$-homotopy category, facilitating new insights into cycle maps.

Findings

Established approximations of Eilenberg-MacLane spaces by smooth projective varieties.

Analyzed the properties of cycle maps from Chow rings to etale cohomology.

Provided tools for studying $A^1$-homotopy types of algebraic 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.