Resumen de Tate and Tate�Hochschild cohomology for finite dimensional Hopf algebras

Van C. Nguyen

• 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 H4.