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.
The author would like to express his appreciation to the referee(s) for valuable suggestions and comments and for pointing out an error in the manuscript. The author also would like to thank Professor Katsunori Sanada, Professor Ayako Itaba and Professor Tomohiro Itagaki for their tremendous support for the improvement of the manuscript of the paper.
"Tate-Hochschild cohomology rings for eventually periodic Gorenstein algebras." SUT J. Math. 57 (2) 133 - 146, December 2021. https://doi.org/10.55937/sut/1641859464