Varieties for modules of finite dimensional Hopf algebras

TL;DR

This paper surveys the theory of support and rank varieties for modules over finite dimensional Hopf algebras, highlighting known properties, examples, and open questions in the field.

Contribution

It provides a comprehensive overview of variety theory for modules over finite dimensional Hopf algebras, emphasizing known results and open problems.

Findings

Support and rank varieties are well-understood for certain classes like finite group algebras.

Open questions remain about tensor products and projectivity related to varieties.

Varieties may help resolve key open problems in module theory for Hopf algebras.

Abstract

We survey variety theory for modules of finite dimensional Hopf algebras, recalling some definitions and basic properties of support and rank varieties where they are known. We focus specifically on properties known for classes of examples such as finite group algebras and finite group schemes. We list open questions about tensor products of modules and projectivity, where varieties may play a role in finding answers.