Support Varieties and Representation Type of Small Quantum Groups

In this paper, we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application, we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild.