Algebraic part of motivic cohomology with compact supports

Abstract

We define an algebraic part for a motivic cohomology group with compact supports of a smooth scheme over an algebraically closed field. This generalizes the classical notion of the algebraic part of a Chow group of a smooth proper variety. We then define algebraic representatives for these algebraic parts as the universal regular homomorphisms with targets in the category of semiabelian varieties. We give a criterion for the existence of a universal regular homomorphism and show the existence in the indices corresponding to, in terms of algebraic cycles, dimension zero and codimensions one and two. (For the codimension one and two cases, we assume that the scheme in question has a smooth compactification with a simple normal crossing boundary divisor.) In dimension zero, our algebraic representative coincides with Serre's generaliezd Albanese variety. We also prove that the algebraic representative in codimension one agrees with the semiabelian variety obtained as the reduction of the identity component of the group scheme that represents the functor of relative Picard groups. This implies that, as in the classical case, the algebraic representative in codimension one is an isomorphism