  Tate-Hochschild cohomology of an algebra is a generalization of ordinary
Hochschild cohomology, which is defined on positive and negative degrees and
has a ring structure. Our purpose of this paper is to study the eventual
periodicity of an algebra by using the Tate-Hochschild cohomology ring. First,
we deal with eventually periodic algebras and show that they are not
necessarily Gorenstein algebras. Secondly, we characterize the eventual
periodicity of a Gorenstein algebra as the existence of an invertible
homogeneous element of the Tate-Hochschild cohomology ring of the algebra,
which is our main result. Finally, we use tensor algebras to establish a way of
constructing eventually periodic Gorenstein algebras.
