Tensor products and support varieties for some noncocommutative Hopf\n algebras

We explore questions of projectivity and tensor products of modules for\nfinite dimensional Hopf algebras. We construct many classes of examples in\nwhich tensor powers of nonprojective modules are projective and tensor products\nof modules in one order are projective but in the other order are not. Our\nexamples are smash coproducts with duals of group algebras, some having algebra\nand coalgebra structures twisted by a cocycle. We develop a theory of support\nvarieties for these Hopf algebras to use as a tool in our investigations.\n