Serre finiteness and Serre vanishing for non-commutative P¹-bundles

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.