Support Varieties and Representation Type of Self-Injective Algebras

We use the theory of varieties for modules arising from Hochschild cohomology to give an alternative version of the wildness criterion of Bergh and Solberg: If a finite dimensional self-injective algebra has a module of complexity at least 3 and satisfies some finiteness assumptions on Hochschild cohomology, then the algebra is wild. We show directly how this is related to the analogous theory for Hopf algebras that we developed. We give applications to many different types of algebras: Hecke algebras, reduced universal enveloping algebras, small half-quantum groups, and Nichols (quantum symmetric) algebras.