Tate-Hochschild cohomology rings for eventually periodic Gorenstein algebras

TL;DR

This paper investigates the Tate-Hochschild cohomology rings of algebras, focusing on their eventual periodicity, and characterizes this property for Gorenstein algebras using invertible elements, also providing construction methods.

Contribution

It characterizes the eventual periodicity of Gorenstein algebras via invertible elements in their Tate-Hochschild cohomology rings and introduces tensor algebra constructions for such algebras.

Findings

Eventually periodic algebras are not necessarily Gorenstein.

Existence of an invertible homogeneous element characterizes Gorenstein algebra periodicity.

Tensor algebras can be used to construct eventually periodic Gorenstein algebras.

Abstract

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.