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