Support varieties and the Hochschild cohomology ring modulo nilpotence

TL;DR

This survey reviews the development of support varieties via Hochschild cohomology for modules over artin algebras and discusses the current understanding of the Hochschild cohomology ring modulo nilpotence.

Contribution

It summarizes the progress and current state of research on the structure of Hochschild cohomology rings modulo nilpotence in the context of support varieties.

Findings

Overview of support varieties theory for artin algebra modules

Summary of research on Hochschild cohomology ring structure

Discussion of open problems and future directions

Abstract

This is a survey paper based on my talks at the 41st Symposium on Ring Theory and Representation Theory, held in Shizuoka University, Japan in September 2008, and will appear in the conference proceedings. The paper begins with a brief introduction to the use of Hochschild cohomology in developing the theory of support varieties for a module over an artin algebra, by Snashall and Solberg (Proc. London Math. Soc.(3) 88 (2004), 705-732). The paper then describes the current status of research concerning the structure of the Hochschild cohomology ring modulo nilpotence.