A relation between Hochschild homology and cohomology for Gorenstein rings

Proc. Amer. Math. Soc. 126 (1998), 1345-1348.

Michel van den Bergh
Departement WNI, Limburgs Universitair Centrum, Universitaire Campus, 
Building D, 3590 Diepenbeek, Belgium 

Abstract. 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 Poincar\'e 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.