A relation between Hochschild homology and cohomology for Gorenstein rings

van den Bergh, Michel

doi:10.1090/S0002-9939-98-04210-5

A relation between Hochschild homology and cohomology for Gorenstein rings
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by Michel van den Bergh

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

DOI: https://doi.org/10.1090/S0002-9939-98-04210-5

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Erratum: Proc. Amer. Math. Soc. 130 (2002), 2809-2810.

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 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.

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