A categorical characterization of quantum projective spaces

Let $R$ be a finite dimensional algebra of finite global dimension over a field $k$. In this paper, we will characterize a $k$-linear abelian category $\mathcal{C}$ such that $\mathcal{C}\cong \mathrm{tails}A$ for some graded right coherent AS-regular algebra $A$ over $R$. As an application, we will prove that if $\mathcal{C}$ is a smooth quadric surface in a quantum ${\mathbb{P}}^3$ in the sense of Smith and Van den Bergh, then there exists a right noetherian AS-regular algebra $A$ over $k{K}_2$ of dimension 3 and of Gorenstein parameter 2 such that $\mathcal{C}\cong \mathrm{tails}A$ where $k{K}_2$ is the path algebra of the 2-Kronecker quiver.