Comparing Hochschild (co)homology for algebras and coalgebras

Question

Given a field $k$, an associative $k$-algebra $A$, and an $A$-bimodule $M$, one can define as the Hochschild homology and cohomology as the homology of the complexes $$M\otimes A^{\otimes n}$$ and $$\text{Hom}(A^{\otimes n},M)$$ respectively (with appropriate differentials)

One can similarly define Hochschild (co)homology for a coassociate coalgebra $C$ and a bicomodule by taking homology of the complexes $$C^{\otimes n}\otimes M$$ and $$\text{Hom}(M, C^{\otimes n})$$

Is it known how the algebra and coalgebra versions are related? In particular, given a Hopf algebra, do both approaches compute the same objects?

Answer 1

We assume $A$ and $M$ are finite dimensional, and denote by $A*$ and $M*$ their respective duals. Denote by $HH_n$ and $CH_n$ the Hochschild homology of an algebra and a coalgebra respectively, and similarly for cohomology. Then a fairly direct computation shows

$$HH_n(A,M)\cong CH_n(M*,A*)$$

and similarly for homology. If $A$ is a Hopf algebra, the two computation give the same object if $A$ is self dual and the bimodule and bicomodule structures on M are dual.