  Let E be a locally free, rank n bimodule over a smooth projective scheme X,
and let A be the non-commutative symmetric algebra generated by E. We construct
an internal Hom functor on the category of graded right A-modules. When E has
rank 2, we prove that A is Gorenstein by computing the right derived functors
of the internal Hom functor. When X is a smooth projective variety, we use the
Gorensteinness of A to prove a version of Serre duality on Proj A, the
non-commutative P^1 bundle defined by A.
