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

Algebraic structure on Tate-Hochschild cohomology of a Frobenius algebra