Van den Bergh, Michel A relation between Hochschild homology and cohomology for Gorenstein rings. (English) Zbl 0894.16005 Proc. Am. Math. Soc. 126, No. 5, 1345-1348 (1998); erratum ibid. 130, No. 9, 2809-2810 (2002). From the 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. Reviewer: W.Soergel (Freiburg i.Br.) Cited in 7 ReviewsCited in 65 Documents MSC: 16E40 (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) Keywords:Hochschild homology; Gorenstein rings; Poincaré duality; enveloping algebras of semisimple Lie algebras PDFBibTeX XMLCite \textit{M. Van den Bergh}, Proc. Am. Math. Soc. 126, No. 5, 1345--1348 (1998; Zbl 0894.16005) Full Text: DOI References: [1] Michael Artin and William F. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), no. 2, 171 – 216. · Zbl 0633.16001 · doi:10.1016/0001-8708(87)90034-X [2] M. Artin, J. Tate, and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, Vol. I, Progr. Math., vol. 86, Birkhäuser Boston, Boston, MA, 1990, pp. 33 – 85. · Zbl 0744.14024 [3] M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), no. 2, 335 – 388. · Zbl 0763.14001 · doi:10.1007/BF01243916 [4] A. Fröhlich, The Picard group of noncommutative rings, in particular of orders, Trans. Amer. Math. Soc. 180 (1973), 1 – 45. · Zbl 0278.16016 [5] C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. · Zbl 0494.16001 [6] Wolfgang Soergel, The Hochschild cohomology ring of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 4, 535 – 538. · Zbl 0871.17005 [7] Michel Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), 1994, pp. 213 – 230. · Zbl 0814.16006 · doi:10.1007/BF00960862 [8] -, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, Journal of Algebra, to appear. · Zbl 0894.16020 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.