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 $\mathscr C$ 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.
Highlights
Let R be a finite dimensional algebra of finite global dimension over a field k
The noncommutative projective scheme associated to an AS-regular algebra of dimension n + 1 is considered as a quantum projective space of dimension n
When is a given k-linear abelian category C equivalent to a quantum projective space? That is, can we find necessary and sufficient conditions on a k-linear abelian category C such that C is equivalent to the noncommutative projective scheme associated to some AS-regular algebra?
Summary
A locally finite N-graded algebra A with A0 = R is called AS-regular over R of dimension d and of Gorenstein parameter l if the following conditions are satisfied:. If R is a Fano algebra, ΠR ∼= B(Db(mod R), R, − ⊗LR ωR−1)≥0 is a graded right coherent AS-regular (Calabi-Yau) algebra of dimension gldim R + 1 and of Gorenstein parameter 1 such that Db(tails ΠR) ∼= Db(mod R) as triangulated categories. A locally finite N-graded algebra A with A0 = R is called ASF-regular of dimension d and of Gorenstein parameter l if the following conditions are satisfied:. If A is an ASF-regular algebra, there exists a graded algebra automorphism ν of A such that D RΓm(A) ∼= Aν(−l)[d] in D(GrMod Ae), so, similar to the connected graded case, we call the graded algebra automorphism ν the (generalized) Nakayama automorphism of A, and we call the graded A-A bimodule ωA := Aν(−l) the canonical module over A (see [12, Section 3.2]). Note that it is conjectured that every AS-regular algebra is graded right coherent
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