Let A be a finite dimensional Hopf algebra over a field k. In this dissertation, we study the Tate cohomology Ĥ* (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 Ĥ* (A, k) is an algebra direct summand of (HH) ̂* (A, A) as a module over Ĥ* (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 Ĥ* (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
