Serre finiteness and Serre vanishing for non-commutative -bundles

Suppose X is a smooth projective scheme of finite type over a field K, is a locally free -bimodule of rank 2, is the non-commutative symmetric algebra generated by and is the corresponding non-commutative -bundle. We use the properties of the internal Hom functor to prove versions of Serre finiteness and Serre vanishing for . As a corollary to Serre finiteness, we prove that is Ext-finite. This fact is used in [I. Mori, J. Pure Appl. Algebra, in press] to prove that if X is a smooth curve over SpecK, has a Riemann–Roch theorem and an adjunction formula.