Tate–Hochschild cohomology for periodic 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.

Characterization of eventually periodic modules in the singularity categories

“…Thanks to Theorem 2.4, we have the following corollary, which generalizes [14, Corollary 6.4], [34,Corollary 3.4] and [33,Theorem 3.5] (in a proper sense).…”