Quantum projective planes and Beilinson algebras of $3$-dimensional quantum polynomial algebras for Type S'

Let $A=\mathcal{A}(E,\sigma)$ be a $3$-dimensional quantum polynomial algebra where $E$ is $\mathbb{P}^{2}$ or a cubic divisor in $\mathbb{P}^{2}$, and $\sigma\in \mathrm{Aut}_{k}E$. Artin-Tate-Van den Bergh proved that $A$ is finite over its center if and only if the order $|\sigma|$ of $\sigma$ is finite. As a categorical analogy of their result, the author and Mori showed that the following conditions are equivalent; (1) $|\nu^{\ast}\sigma^{3}|<\infty$, where $\nu$ is the Nakayama automorphism of $A$. (2) The norm $\|\sigma\|$ of $\sigma$ is finite. (3) The quantum projective plane $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. In this paper, we will prove for Type S' algebra $A$ that the following conditions are equivalent; (1) $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. (2) The Beilinson algebra $\nabla A$ of $A$ is $2$-representation tame. (3) The isomorphism classes of simple $2$-regular modules over $\nabla A$ are parametrized by $\mathbb{P}^{2}$.