  We study cup product and cap product in Tate-Hochschild theory for a finite
dimensional Frobenius algebra. We show that Tate-Hochschild cohomology ring
equipped with cup product is isomorphic to singular Hochschild cohomology ring
introduced by Wang. An application of cap product occurs in Tate-Hochschild
duality; as in Tate (co)homology of a finite group, the cap product with the
fundamental class of a finite dimensional Frobenius algebra provides certain
duality result between Tate-Hochschild cohomology and homology groups.
Moreover, we characterize minimal complete resolutions over a finite
dimensional self-injective algebra by means of the notion of minimal complexes
introduced by Avramov and Martsinkovsky.

Representation Theory;Rings and Algebras