Serre duality for non-commutative ${\mathbb{P}}^{1}$-bundles

Let $X$ be a smooth scheme of finite type over a field $K$, let $\mathcal{E}$ be a locally free $\mathcal{O}_{X}$-bimodule of rank $n$, and let $\mathcal{A}$ be the non-commutative symmetric algebra generated by $\mathcal{E}$. We construct an internal $\operatorname{Hom}$ functor, ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (-,-)$, on the category of graded right $\mathcal{A}$-modules. When $\mathcal{E}$ has rank 2, we prove that $\mathcal{A}$ is Gorenstein by computing the right derived functors of ${\underline{{\mathcal{H}}\textit{om}}_{\mathsf{Gr} \mathcal{A}}} (\mathcal{O}_{X},-)$. When $X$ is a smooth projective variety, we prove a version of Serre Duality for ${\mathsf{Proj}} \mathcal{A}$ using the right derived functors of $\underset{n \to \infty}{\lim} \underline{\mathcal{H}\textit{om}}_{\mathsf{Gr} \mathcal{A}} (\mathcal{A}/\mathcal{A}_{\geq n}, -)$.