Bivariant cyclic cohomology and Connes’ bilinear pairings in noncommutative motives

Abstract

Abstract. In this article we further the study of non-commutative motives, initiated in [3,4,25]. We prove that bivariant cyclic cohomology (and its variants) becomes representable in the category Mot loc dg (e) of non-commutative motives. Furthermore, Connes' bilinear pairings correspond to the composition operation in Mot loc dg (e). As an application, we obtain a simple model, given in terms of infinite matrices, for the (de)suspension of these bivariant cohomology theories.