Products, multiplicative Chern characters, and finite coefficients via Non-commutative motives

TL;DR

This paper uses non-commutative motives to provide a conceptual understanding of products, Chern characters, and finite coefficients in algebraic K-theory, revealing their universal properties and extending classical constructions.

Contribution

It characterizes key algebraic K-theory tools via universal properties using non-commutative motives, simplifying their understanding and extending classical results.

Findings

Multiplicativity of negative Chern characters derived from a simple factorization.

Extension of Kassel's bivariant Chern character to higher algebraic K-theory.

Universal properties clarify the conceptual framework of algebraic K-theory tools.

Abstract

Products, multiplicative Chern characters, and finite coefficients, are unarguably among the most important tools in algebraic K-theory. Although they admit numerous different constructions, they are not yet fully understood at the conceptual level. In this article, making use of the theory of non-commutative motives, we change this state of affairs by characterizing these constructions in terms of simple, elegant, and precise universal properties. We illustrate the potential of our results by developing two of its manyfold consequences: (1) the multiplicativity of the negative Chern characters follows directly from a simple factorization of the mixed complex construction; (2) Kassel's bivariant Chern character admits an adequate extension, from the Grothendieck group level, to all higher algebraic K-theory.