Regulators on additive higher Chow groups

TL;DR

This paper introduces cubical additive higher Chow groups with modulus to explore motives over certain algebraic structures, providing explicit regulator maps for 1-cycles using advanced residue theory.

Contribution

It extends the theory of additive higher Chow groups to all dimensions and constructs explicit regulator maps for 1-cycles.

Findings

Defined cubical additive higher Chow groups with modulus for all dimensions.

Constructed explicit regulator maps on 1-cycle groups.

Extended previous work on 0-dimensional cycles to higher dimensions.

Abstract

As an attempt to understand motives over $k[x]/(x^m)$, we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. R\"ulling on 0-dimensional cycles. We give an explicit construction of regulator maps on the groups of 1-cycles with an aid of the residue theory of A. Parshin and V. Lomadze.