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

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.