A categorical characterization of quantum projective spaces

Abstract

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.

31 pages, v2: The proof of Theorem 3.10 of the first version was not correct. Accordingly, the statement of the main result (Theorem 4.1) has been revised, v3 and v4: minor revision

References in corpus (2)

• A tour about existence and uniqueness of dg enhancements and lifts
• Cluster tilting modules and noncommutative projective schemes

Cited by in corpus (2)

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