Relation between Hochschild homology and cohomology for Gorenstein rings

Abstract

Let "HH" stand for Hochschild (co)homology. In this note we show that for many rings A there exists d is an element of N such that for an arbitrary A-bimodule N we have HHi(N) = HHd-z(N). Such a result may be viewed as an analog of Poincare 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.