Bivariant cyclic cohomology and Connes’ bilinear pairings in noncommutative motives

In this article we further the study of noncommutative motives. We prove that the bivariant cohomology and the bivariant Chern character of any additive invariant E become representable in the category of noncommutative motives. This applies in particular to bivariant cyclic cohomology and its variants. When E is moreover symmetric monoidal we prove that the associated Chern character is multiplicative and characterize it by a precise universal property. In the particular case of bivariant cyclic cohomology the associated Chern character becomes the universal lift of the Dennis trace map. Then, we prove that under the above representability result, the composition operation in the category of noncommutative motives identifies with Connes’ bilinear pairings. As an application, we obtain a simple model, given by Karoubi’s infinite matrices, for the (de)suspension of these bivariant cohomology theories.