Algebraic part of motivic cohomology with compact supports

TL;DR

This paper extends the concept of algebraic equivalence and regular homomorphisms to Voevodsky motives, establishing the existence of universal regular homomorphisms for certain motivic cohomology groups, generalizing classical results.

Contribution

It generalizes Murre's work to Voevodsky motives, proving the existence of universal regular homomorphisms in the Nisnevich topology for a broad class of motivic cohomology groups.

Findings

Existence of universal regular homomorphisms in Nisnevich topology

Recovery of Murre's theorem and classical cases

Applicability to higher Chow groups and Milnor K-groups

Abstract

Motivated by Murre's work on universal regular homomorphisms on Chow groups in codimension $2,$ we generalize the algebraic equivalence relation and regular homomorphisms to the context of Voevodsky motives over a field. In the Nisnevich topology, we prove the existence of \emph{universal} regular homomorphisms for a certain class of motivic cohomology groups, recovering Murre's theorem and the existence of Picard and Albanese varieties as special cases. This class also includes interesting cases such as higher Chow groups and Milnor $K$-groups. The appendix by Kahn proves that, for \'etale motives, universal regular homomorphisms exist for all geometric motives and compares them with those in the Nisnevich topology when both exist.