Products in Hopf-Cyclic Cohomology

TL;DR

This paper develops new pairings in Hopf-cyclic cohomology using derived functor and Yoneda interpretations, recovering known characteristic maps and demonstrating their invariance under various categorical replacements.

Contribution

It introduces several new pairings in Hopf-cyclic cohomology based on derived functor and Yoneda methods, extending the understanding of invariance in these structures.

Findings

Recovered the Connes-Moscovici characteristic map as a special pairing.

Established invariance of pairings under categorical replacements.

Extended the framework of Hopf-cyclic cohomology pairings.

Abstract

We construct several pairings in Hopf-cyclic cohomology of (co)module (co)algebras with arbitrary coefficients. The key ideas instrumental in constructing these pairings are the derived functor interpretation of Hopf-cyclic and equivariant cyclic (co)homology, and the Yoneda interpretation of Ext-groups. As a special case of one of these pairings, we recover the Connes-Moscovici characteristic map in Hopf-cyclic cohomology. We also prove that this particular pairing, along with few others, would stay the same if we replace the derived category of (co)cyclic modules with the homotopy category of (special) towers of $X$-complexes, or the derived category of mixed complexes.