Hochschild duality, localization and smash products

Abstract

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 $V$.