Voevodsky’s motives and Weil reciprocity

Abstract

We describe Somekawa’s $K$-group associated to a finite collection of semiabelian varieties (or more general sheaves) in terms of the tensor product in Voevodsky’s category of motives. While Somekawa’s definition is based on Weil reciprocity, Voevodsky’s category is based on homotopy invariance. We apply this to explicit descriptions of certain algebraic cycles.