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A relation between Hochschild homology
and cohomology for Gorenstein rings


Author: Michel van den Bergh
Journal: Proc. Amer. Math. Soc. 126 (1998), 1345-1348
MSC (1991): Primary 16E40
DOI: https://doi.org/10.1090/S0002-9939-98-04210-5
Erratum: Proc. Amer. Math. Soc. 130 (2002), 2809-2810.
MathSciNet review: 1443171
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Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), 171-216. MR 88k:16003
  • 2. M. Artin, J. Tate, and M. van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift, vol. 1, Birkhäuser, 1990, pp. 33-85. MR 92e:14002
  • 3. -, Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), 335-388. MR 93e:16055
  • 4. A. Fröhlich, The Picard group of non-commutative rings, Trans. Amer. Math. Soc. 180 (1973), 1-45. MR 47:6751
  • 5. C. Nastacescu and F. Van Oystaeyen, Graded ring theory, North-Holland, 1982. MR 84i:16002
  • 6. W. Soergel, The Hochschild cohomology of regular maximal primitive quotients of enveloping algebras of semisimple Lie algebras, Ann. Sci. École Norm. Sup. (4) 29 (1996), 535-538. MR 97e:17016
  • 7. M. van den Bergh, Non-commutative homology of some three dimensional quantum spaces, J. K-theory (1994), 213-230. MR 95i:16009
  • 8. -, Existence theorems for dualizing complexes over non-commutative graded and filtered rings, Journal of Algebra, to appear.

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Additional Information

Michel van den Bergh
Affiliation: Departement WNI, Limburgs Universitair Centrum, Universitaire Campus, Building D, 3590 Diepenbeek, Belgium
Email: vdbergh@luc.ac.be

DOI: https://doi.org/10.1090/S0002-9939-98-04210-5
Keywords: Hochschild homology, Gorenstein rings
Received by editor(s): November 5, 1996
Additional Notes: The author is a senior researcher at the NFWO
Communicated by: Lance W. Small
Article copyright: © Copyright 1998 American Mathematical Society

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