The cyclic theory of Hopf algebroids

TL;DR

This paper develops a comprehensive cyclic cohomology and homology framework for Hopf algebroids, linking module categories and duality, with applications to Lie-Rinehart algebras and étale groupoids.

Contribution

It introduces a dual cyclic homology theory and provides structure theorems for commutative and cocommutative cases, with explicit computations in key examples.

Findings

Established a systematic cyclic cohomology description for Hopf algebroids

Introduced a dual cyclic homology theory via cyclic duality

Computed cyclic theories for Lie-Rinehart algebras and étale groupoids

Abstract

We give a systematic description of the cyclic cohomology theory of Hopf algebroids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying cyclic duality to the underlying cocyclic object. We derive general structure theorems for these theories in the special cases of commutative and cocommutative Hopf algebroids. Finally, we compute the cyclic theory in examples associated to Lie-Rinehart algebras and \'etale groupoids.