Rank Varieties for Hopf Algebras

TL;DR

This paper introduces rank varieties for specific Hopf algebras, linking them to cohomological support varieties and establishing finite generation of Ext groups, advancing the understanding of module theory in Hopf algebra contexts.

Contribution

It constructs rank varieties for the Drinfel'd double of the Taft algebra and U_q(sl2), connecting them to cohomological support varieties and demonstrating Ext^* finiteness.

Findings

Rank varieties are homeomorphic to cohomological support varieties for certain Hopf algebras.

Ext^*(M,M) is finitely generated over the cohomology ring for finitely-generated modules.

The work extends the theory of support varieties to new classes of Hopf algebras.

Abstract

We construct rank varieties for the Drinfel'd double of the Taft algebra and for U_q(sl2). For the Drinfel'd double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of A-modules whenever A is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext^*(M,M) is finitely generated over the cohomology ring of the Drinfel'd double for any finitely-generated module M.