Multiplicative properties of the current transform regulator

TL;DR

This paper explores the multiplicative properties of a regulator map derived from the transform of currents, using advanced algebraic and categorical techniques to establish its group-like behavior.

Contribution

It introduces a synthetic approach to fundamental triples of currents and demonstrates their multiplicative nature within the derived category framework.

Findings

Establishes the multiplicative property of the regulator map.

Shows group-like behavior of currents under an extended Eilenberg-Zilber morphism.

Identifies a character of the permutation Hopf algebra with values in an infinite-dimensional function field.

Abstract

This paper utilizes the properties of transforms of currents under equidimensional cycles, as introduced in \cite{MR4498559}, to establish the multiplicative nature of the resulting regulator map, in the derived category. The construction relies on a synthetic presentation of the fundamental triples of currents from \cite{MR4498559}, which exhibits group-like behavior under an extended Eilenberg-Zilber morphism. A key component of the analysis is a character of the permutation Hopf algebra that takes values in the function field of an infinite-dimensional affine space.