Voevodsky's motives and Weil reciprocity

TL;DR

This paper connects Somekawa's K-groups, defined via Weil reciprocity, with Voevodsky's motivic category based on homotopy invariance, enabling explicit descriptions of algebraic cycles.

Contribution

It provides a new interpretation of Somekawa's K-groups within Voevodsky's motivic framework, bridging two different approaches.

Findings

Expressed Somekawa's K-groups in terms of Voevodsky's motives

Linked Weil reciprocity with homotopy invariance in motives

Enabled explicit descriptions of algebraic cycles

Abstract

We describe Somekawa's K-group associated to a finite collection of semi-abelian 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.