  In this work we study the class of algebras satisfying a duality property
with respect to Hochschild homology and cohomology, as in [VdB]. More
precisely, we consider the class of algebras $A$ such that there exists an
invertible bimodule $U$ and an integer number $d$ with the property
$H^{\bullet}(A,M)\cong H_{d-\bullet}(A,U\ot_AM)$, for all $A$-bimodules $M$. We
will show that this class is closed under localization and under smash products
with respect to Hopf algebras satisfying also the duality property. We also
illustrate the subtlety on dualities with smash products developing in detail
the example $S(V)#G$, the crossed product of the symmetric algebra on a vector
space and a finite group acting linearly on $V$.
