We review recent progress in the study of cyclic cohomology of Hopf algebras, Hopf algebroids, and invariant cyclic homology starting with the pioneering work of Connes-Moscovici.

Generalizing work of Doi and of Farinati and Solotar, we define coHochschild and cocyclic homology theories for chain coalgebras over any commutative ring and prove their naturality with respect to morphisms of chain coalgebras up to strong homotopy. As a consequence we obtain that if the comultiplication of a chain coalgebra is itself a morphism of chain coalgebras up to strong homotopy, then ...

We identify the periodic cyclic homology of the algebra of complete symbols on a differential groupoid G in terms of the cohomology of S * (G), the cosphere bundle of A(G), where A(G) is the Lie algebroid of G. We also relate the Hochschild homology of this algebra with the homogeneous Poisson homology of the space A * (G) 0 ∼ = S * (G) × (0, ∞), the dual of A(G) with the zero section removed. ...

The central result here is an explicit computation of the Hochschild and cyclic homologies of a natural smooth subalgebra of stable continuous trace algebras having smooth manifolds X as their spectrum. More precisely, the Hochschild homology is identified with the space of differential forms on X, and the periodic cyclic homology with the twisted de Rham cohomology of X, thereby generalizing s...

Strong flatness properties are established for a large class of functional-analytic rings including all C*-algebras. This is later used to prove that all those rings satisfy excision in Hochschild and in cyclic homology over almost arbitrary rings of coefficients and that, for stable C*-algebras, the Hochschild and cyclic homology groups defined over an arbitrary coefficient ring k subset C of ...

Abstract. In this paper, we introduce a Z-graded variant of the periodic cyclic homology of associative algebras which generalizes the infinitesimal cohomology of affine algebras in characteristic 0 and show that it satisfies the K0nneth formula (i.e. it commutes with the formation of tensor products). We also show that the Kllnneth formula in periodic cyclic homology holds only under the prese...