Bivariant cyclic cohomology and Connes' bilinear pairings in Non-commutative motives

TL;DR

This paper advances the understanding of non-commutative motives by proving bivariant cyclic cohomology's representability and linking Connes' pairings to composition, with applications to models using infinite matrices.

Contribution

It demonstrates the representability of bivariant cyclic cohomology in non-commutative motives and relates Connes' pairings to composition operations.

Findings

Bivariant cyclic cohomology is representable in non-commutative motives.

Connes' bilinear pairings correspond to composition operations.

Provides a matrix-based model for (de)suspension of bivariant cohomology theories.

Abstract

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