Let A be a finite dimensional Hopf algebra over a field k. In this dissertation, we study the Tate cohomology ?* (A, k) and Tate-Hochschild cohomology (HH) ?* (A, A) of A, and their properties. We introduce cup products that make them become graded-commutative rings and establish the relationship between these rings. In particular, we show ?* (A, k) is an algebra direct summand of (HH) ?* (A, A) as a module over ?* (A, k). When A is a finite group algebra RG over a commutative ring R, we show that the Tate-Hochschild cohomology ring (HH) ?* (RG, RG) of RG is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of G. Moreover, our main result provides an explicit formula for the cup product in (HH) ?* (RG, RG) with respect to this decomposition. When A is symmetric, we show that there are finitely generated A-modules whose Tate cohomology is not finitely generated over the Tate cohomology ring ?* (A, k) of A. It turns out that if a module in a connected component of the stable Auslander-Reiten quiver associated to A has finitely generated Tate cohomology, then so does every module in that component.

Let A be a finite dimensional Hopf algebra over a field k. In this dissertation, we study the Tate cohomology ?* (A, k) and Tate-Hochschild cohomology (HH) ?* (A, A) of A, and their properties. We introduce cup products that make them become graded-commutative rings and establish the relationship between these rings. In particular, we show ?* (A, k) is an algebra direct summand of (HH) ?* (A, A) as a module over ?* (A, k).

When A is a finite group algebra RG over a commutative ring R, we show that the Tate-Hochschild cohomology ring (HH) ?* (RG, RG) of RG is isomorphic to a direct sum of the Tate cohomology rings of the centralizers of conjugacy class representatives of G. Moreover, our main result provides an explicit formula for the cup product in (HH) ?* (RG, RG) with respect to this decomposition.

When A is symmetric, we show that there are finitely generated A-modules whose Tate cohomology is not finitely generated over the Tate cohomology ring ?* (A, k) of A. It turns out that if a module in a connected component of the stable Auslander-Reiten quiver associated to A has finitely generated Tate cohomology, then so does every module in that component.