Tate and Tate-Hochschild Cohomology for finite dimensional Hopf Algebras
  Let A be any finite dimensional Hopf algebra over a field k. We specify the Tate and Tate-Hochschild cohomology for A and introduce cup products that make them become graded rings. We establish the relationship between these rings. In particular, the Tate-Hochschild cohomology of A is isomorphic (as algebras) to its Tate cohomology with coefficients in an adjoint module. Consequently, the Tate cohomology ring of A is a direct summand of its Tate-Hochschild cohomology ring. As an example, we explicitly compute both the Tate and Tate-Hochschild cohomology for the Sweedler algebra H_4. 
Van C. Nguyen
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