Tate and Tate-Hochschild Cohomology for finite dimensional Hopf Algebras

Abstract

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.

arXiv comment18 pages, simplified from the first version, to appear in J. Pure and Applied Algebra

References in corpus (1)

• Hochschild cohomology of smash products and rank one Hopf algebras

Cited by in corpus (2)

• The Tate-Hochschild cohomology ring of a group algebra
• Algebraic structure on Tate-Hochschild cohomology of a Frobenius algebra