A relation between Hochschild homology and cohomology for Gorenstein rings

Let ``$HH$'' stand for Hochschild (co)homology. In this note we show that for many rings $A$ there exists $d\in {\mathbb N}$ such that for an arbitrary $A$-bimodule $N$ we have $HH^i(N)=HH_{d-i}(N)$. Such a result may be viewed as an analog of Poincaré duality.

Combining this equality with a computation of Soergel allows one to compute the Hochschild homology of a regular minimal primitive quotient of an enveloping algebra of a semisimple Lie algebra, answering a question of Polo.