JournalsjncgVol. 7, No. 4pp. 907–937

Tate-Hochschild homology and cohomology of Frobenius algebras

  • Petter Andreas Bergh

    NTNU Trondheim, Norway
  • David A. Jorgensen

    University of Texas at Arlington, USA
Tate-Hochschild homology and cohomology of Frobenius algebras cover
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Abstract

Let Λ\Lambda be a two-sided Noetherian Gorenstein kk-algebra, for kk a field. We introduce Tate–Hochschild homology and cohomology groups for Λ\Lambda, which are defined for all degrees, non-negative as well as negative, and which agree with the usual Hochschild homology and cohomology groups for all degrees larger than the injective dimension of Λ\Lambda. We prove certain duality theorems relating the Tate–Hochschild (co)homology groups in positive degree to those in negative degree, in the case where Λ\Lambda is a Frobenius algebra. We explicitly compute all Tate–Hochschild (co)homology groups for certain classes of Frobenius algebras, namely, certain quantum complete intersections.

Cite this article

Petter Andreas Bergh, David A. Jorgensen, Tate-Hochschild homology and cohomology of Frobenius algebras. J. Noncommut. Geom. 7 (2013), no. 4, pp. 907–937

DOI 10.4171/JNCG/139