Tate and Tate-Hochschild Cohomology for finite dimensional Hopf Algebras

TL;DR

This paper develops the theory of Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras, establishing their algebraic structures and relationships, with explicit computations for the Sweedler algebra.

Contribution

It introduces graded ring structures on Tate and Tate-Hochschild cohomology for finite dimensional Hopf algebras and relates these cohomologies, including explicit calculations for a specific algebra.

Findings

Tate and Tate-Hochschild cohomologies are graded rings.

Tate-Hochschild cohomology is isomorphic to Tate cohomology with an adjoint module.

Explicit cohomology computations for the Sweedler algebra H_4.

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.