Rank varieties for Hopf algebras

We construct rank varieties for the Drinfeld double of the Taft algebra Λn and for uq(sl2). For the Drinfeld double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of Λ-modules whenever Λ 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 Drinfeld double for any finitely generated module M.