Cyclic homology

For an algebra A over a unitary commutative ring k, we have the Hochschild homology HH_*(A).  One use of it was a generalization of the modules of differential forms to non-commutative algebras.  This gave us HDR_*(A), the non-commutative de Rham homology, developed by Alain Connes in his paper “Non-commutative differential geometry.”  In that paper he also produced cyclic homology, HC_*(A), which is connected to both Hochschild and de Rham homology.  The nicest connection between them is when k contains Q.  Then we get the Karoubi exact sequence

0 —> HDR_n(A) —> HC_n(A) —> HH_{n+1}(A).

In the study of quantum groups, cyclic homology is generalized to twisted cyclic homology for a pair of an algebra together with a given k-algebra automorphism.  I was able to extend Karoubi’s theorem to twisted cyclic homology and also to twisted cyclic homology for crossed product algebras (an algebra together with a group of k-algebra automorphisms).  Another extension of cyclic homology is to coalgebras, producing a cyclic cohomology.  One example would be the coalgebra of a Frobenius algebra.