A relation between Hochschild homology and cohomology for Gorenstein rings

Let “ H H HH ” stand for Hochschild (co) homology. In this note we show that for many rings A A there exists d ∈ N d N such that for an arbitrary A A -bimodule N N we have H H i (N) = H H d − i (N) HHⁱ (N) =HH₃-₈ (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.