A universal characterization of the Chern character maps

TL;DR

This paper provides a universal characterization of the Chern character maps, clarifying their conceptual understanding by defining them through simple universal properties, which enhances their theoretical foundation.

Contribution

It introduces a novel universal property-based framework for understanding Chern character maps from the Grothendieck group to cyclic homology.

Findings

Chern character maps are characterized by universal properties.

The approach unifies various constructions of Chern characters.

Provides a clearer conceptual understanding of Chern characters.

Abstract

The Chern character maps are one of the most important working tools in mathematics. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this note we eliminate this gap by characterizing the Chern character maps, from the Grothendieck group to the (negative) cyclic homology groups, in terms of simple universal properties.