Products, multiplicative Chern characters, and finite coefficients via noncommutative motives

Products, multiplicative Chern characters, and finite coefficients, are unarguably among the most important tools in algebraic K-theory. In this article, making use of the theory of noncommutative motives, we characterize these constructions in terms of simple and precise universal properties. We illustrate the potential of these results by developing two of its manifold 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 K-theory groups.