Let be a two-sided Noetherian Gorenstein -algebra, for a field. We introduce Tate–Hochschild homology and cohomology groups for , 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 . We prove certain duality theorems relating the Tate–Hochschild (co)homology groups in positive degree to those in negative degree, in the case where 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. 907–937DOI 10.4171/JNCG/139