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Let $A=\mathcal{A}(E,\sigma)$ be a $3$-dimensional quantum polynomial algebrawhere $E$ 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 $A$ isfinite over its center if and only if the order $|\sigma|$ of $\sigma$ isfinite. As a categorical analogy of their result, the author and Mori showedthat 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 quantumprojective plane $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. In thispaper, we will prove for Type S' algebra $A$ that the following conditions areequivalent; (1) $\mathsf{Proj}_{{\rm nc}}A$ is finite over its center. (2) TheBeilinson algebra $\nabla A$ of $A$ is $2$-representation tame. (3) Theisomorphism classes of simple $2$-regular modules over $\nabla A$ areparametrized by $\mathbb{P}^{2}$.

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