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