Morita base change in Hopf-cyclic (co)homology

TL;DR

This paper proves that cyclic (co)homology remains invariant under Morita base change for left Hopf algebroids, extending classical results and applying to noncommutative tori.

Contribution

It establishes Morita invariance of cyclic (co)homology for left Hopf algebroids and applies this to noncommutative tori, linking to Lie algebroid homology.

Findings

Cyclic (co)homology is invariant under Morita base change.

Classical Morita invariance extends to Hopf algebroids.

Homology of noncommutative tori relates to Lie algebroid homology.

Abstract

In this paper, we establish the invariance of cyclic (co)homology of left Hopf algebroids under the change of Morita equivalent base algebras. The classical result on Morita invariance for cyclic homology of associative algebras appears as a special example of this theory. In our main application we consider the Morita equivalence between the algebra of complex-valued smooth functions on the classical 2-torus and the coordinate algebra of the noncommutative 2-torus with rational parameter. We then construct a Morita base change left Hopf algebroid over this noncommutative 2-torus and show that its cyclic (co)homology can be computed by means of the homology of the Lie algebroid of vector fields on the classical 2-torus.