  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
$\mathscr C$ such that $\mathscr C\cong \operatorname {tails} A$ for some
graded right coherent AS-regular algebra $A$ over $R$. As an application, we
will prove that if $\mathscr C$ 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 $kK_2$ of dimension 3 and of
Gorenstein parameter 2 such that $\mathscr C\cong \operatorname {tails} A$
where $kK_2$ is the path algebra of the 2-Kronecker quiver.
