  Suppose $X$ is a smooth projective scheme of finite type over a field $K$,
$\mathcal{E}$ is a locally free ${\mathcal{O}}_{X}$-bimodule of rank 2,
$\mathcal{A}$ is the non-commutative symmetric algebra generated by
$\mathcal{E}$ and ${\sf Proj}\A$ is the corresponding non-commutative
$\mathbb{P}^{1}$-bundle. We use the properties of the internal
$\operatorname{Hom}$ functor $\HU(-,-)$ to prove versions of Serre finiteness
and Serre vanishing for ${\sf Proj}\A$. As a corollary to Serre finiteness, we
prove that ${\sf Proj}\A$ is Ext-finite. This fact is used in \cite{izu} to
prove that if $X$ is a smooth curve over $\operatorname{Spec}K$, ${\sf Proj
}\A$ has a Riemann-Roch theorem and an adjunction formula.
