arXiv:1109.4019 Tate-Hochschild homology and cohomology of Frobenius algebras from arXiv Front: math.KT by Petter Andreas Bergh, David A. Jorgensen 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.
arXiv:1209.4888 Tate and Tate-Hochschild Cohomology for finite dimensional Hopf Algebras from arXiv Front: math.KT by Van C. Nguyen Let A be any finite dimensional Hopf algebra over a field k. We generalize the notion of Tate cohomology for A, which is defined in both positive and negative degrees, and compare it with the Tate-Hochschild cohomology of A that was presented by Bergh and Jorgensen. We introduce cup products that make the Tate and Tate-Hochschild cohomology of A become graded rings. We establish the relationship between these rings, which turns out to be similar to that in the ordinary non-Tate cohomology case. As an example, we explicitly compute the Tate-Hochschild cohomology for a finite dimensional (cyclic) group algebra. In another example, we compute both the Tate and Tate-Hochschild cohomology for a Taft algebra, in particular, the Sweedler algebra H_4.
nLab page on Tate-Hochschild cohomology