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 $\mathscr C$ 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.

introduced and studied the cohomology groups of coherent sheaves on projective schemes.In particular, he proved that the category coh(Proj A) of coherent sheaves on the projective scheme Proj A is equivalent to the category of finitely generated graded A-modules modulo finite dimensional modules.In 1994, motivated by Serre's work, M. Artin and J. J. Zhang [1] introduced the categorical notion of a noncommutative projective scheme, and established a fundamental theory of noncommutative projective schemes.Since then, the study of noncommutative projective schemes has been one of the major projects in noncommutative algebraic geometry.
The noncommutative projective scheme associated to an AS-regular algebra of dimension n + 1 is considered as a quantum projective space of dimension n.Since projective spaces are the most basic and important class of projective schemes in commutative algebraic geometry, quantum projective spaces have been studied deeply and extensively in noncommutative algebraic geometry.It is known that although quantum projective spaces have many nice properties in common with coh P n , their structures vary widely.In this paper, we consider the following question.
Question 1.1.Fix a field k.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?
If a k-linear abelian category C is equivalent to a quantum projective space, then we can investigate C using the rich techniques of noncommutative algebraic geometry.In this sense, Question 1.1 is important.The following is the main result of this paper, which gives a complete answer to Question 1.1.
Theorem 1.2 (Theorem 4.1).Let R be a finite dimensional algebra of finite global dimension over a field k.Then a k-linear abelian category C is equivalent to the noncommutative projective scheme associated to some AS-regular algebra A over A 0 ∼ = R of Gorenstein parameter ℓ if and only if (AS1) C has a canonical bimodule ω C , and (AS2) there exist an object O ∈ C and a k-linear autoequivalence s ∈ Aut k C such that (a) (O, s) is ample for C (in the sense of Artin and Zhang [1]), (b) {s i O} i∈Z is a full geometric relative helix of period ℓ for D b (C ), and (c) End C (O) ∼ = R.
Roughly speaking, (AS1) requires that C has an autoequivalence which induces a Serre functor for D b (C ) (Definition 3.4), and (AS2)(b) requires that D b (C ) has a "relaxed" version of a full geometric helix, consisting of shifts of a single object in C (Definitions 3.12, 3.14).
In the last section, we will give an application of the main result.It is well-known that if Q is a smooth quadric surface in P 3 , then there exists a noetherian AS-regular algebra A = k x, y /(x 2 y − yx 2 , xy 2 − y 2 x) of dimension 3 such that coh Q is equivalent to the noncommutative projective scheme associated to A. Using our main result, we will prove a noncommutative generalization of this result.Namely we will show that if C is a smooth quadric surface in a quantum P 3 in the sense of Smith and Van den Bergh [19], then there exists a right noetherian AS-regular algebra A over kK 2 of dimension 3 such that C is equivalent to the noncommutative projective scheme associated to A where kK 2 is the path algebra of the 2-Kronecker quiver K 2 (Theorem 5.17).
1.2.Notation.In this subsection, we introduce some notation and terminology that will be used in this paper.Throughout, let k be a field.We assume that all algebras are over k.For an algebra R, we denote by Mod R the category of right R-modules, and by mod R the full subcategory consisting of finitely presented right R-modules.Note that if R is a finite dimensional algebra, then mod R is simply the full subcategory consisting of finite dimensional R-modules.We denote by R o the opposite algebra of R and define R e := R o ⊗ k R. For algebras R, S, Mod R o is identified with the category of left Rmodules, and Mod(R o ⊗ k S) is identified with the category of R-S bimodules, so that Mod R e is identified with the category of R-R bimodules.
For a vector space V over k, we denote by DV = Hom k (V, k) the vector space dual of V over k.By abuse of notation, for a graded vector space V = i∈Z V i , we denote by DV the graded vector space dual of V defined as (DV ) i = D(V −i ) for i ∈ Z.We say that a graded vector space V is locally finite if dim k V i < ∞ for all i ∈ Z.In this case, we define the Hilbert series of V by In this paper, a graded algebra means a Z-graded algebra over a field k, although we mainly deal with N-graded algebras.For a graded algebra A, we denote by GrMod A the category of graded right A-modules, and by grmod A the full subcategory consisting of finitely presented graded right A-modules.Morphisms in GrMod A are A-module homomorphisms preserving degrees.For M ∈ GrMod A and a graded algebra automorphism σ of A, we define the twist M σ ∈ GrMod A by M σ = M as a graded k-vector space with the new right action m * a = mσ(a).
Let A be a Z-graded algebra, and r ∈ N + .The r-th Veronese algebra of A is defined by and the r-th quasi-Veronese algebra of A is defined by where the multiplication of A [r] is given by (a ij )(b ij ) = ( k a kj b ik ) (see [13]).There exists an equivalence functor Q : GrMod A → GrMod A [r] defined by where the right action of A [r] on Q(M) is given by (m i )(a ij ) = ( k m k a ik ) (see [13]).Let A = i∈N A i be an N-graded algebra.We say that A is connected graded if A 0 = k.For a graded module M ∈ GrMod A and an integer n ∈ Z, we define the truncation M ≥n := i≥n M i ∈ GrMod A and the shift M(n) ∈ GrMod A by M(n) i := M n+i for i ∈ Z.The rule M → M(n) is a k-linear autoequivalence for GrMod A, called the shift functor.For M, N ∈ GrMod A, we write the vector space Ext i A (M, N) := Ext i GrMod A (M, N) and the graded vector space Ext Let A, C be N-graded algebras.Then C o ⊗ k A becomes an N-graded algebra by setting where m = A ≥1 .The derived functor of Γ m is denoted by RΓ m , and its cohomologies are denoted by We say that A has finite cohomological dimension if cd(Γ m ) < ∞.Note that if A has finite global dimension, then it has finite cohomological dimension.
Connected graded AS-regular algebras defined below are the most important class of algebras in noncommutative algebraic geometry.Definition 1.3.A locally finite connected graded algebra A is called AS-regular (resp.AS-Gorenstein) of dimension d and of Gorenstein parameter ℓ if the following conditions are satisfied: (1 It is well-known that if A is a noetherian AS-Gorenstein algebra of dimension d and of Gorenstein parameter ℓ, then A has a balanced dualizing complex D RΓ m (A) ∼ = A ν (−ℓ) [d] in D(GrMod A e ) with some graded algebra automorphism ν of A (see [24]).This graded algebra automorphism ν is called the (generalized) Nakayama automorphism of A. The graded A-A bimodule ω A := A ν (−ℓ) ∈ GrMod A e is called the canonical module over A.
Let us recall the definition of graded coherentness.
(1) A graded right A-module M is called graded right coherent if it is finitely generated and every finitely generated graded submodule of M is finitely presented over A.
(2) A locally finite N-graded algebra A is called graded right coherent if A and A/A ≥1 are graded right coherent modules.
Let A be a graded right coherent algebra.Then a graded right A-module is finitely presented if and only if it is graded right coherent.In this case, grmod A is an abelian category.
Proposition 1.5 (cf.[24,Proposition 1.9]).If A is a graded right coherent algebra, then every finite dimensional graded right A-module is graded right coherent.
Proof.If S is a graded simple right A-module, then there exists a surjection A/A ≥1 (j) → S for some j ∈ Z.Let K be the kernel of this map.Since it is finite dimensional, it is a finitely generated submodule of A/A ≥1 (j).Since A/A ≥1 (j) is graded right coherent, K is finitely presented.Since grmod A is an abelian category, S is graded right coherent.Since every finite dimensional module is a finite extension of graded simple right A-modules, the result follows.
Let A be a graded right coherent algebra.We denote by tors A the full subcategory of grmod A consisting of finite dimensional modules.By Proposition 1.5, tors A is a Serre subcategory of grmod A, so the quotient category tails A := grmod A/ tors A is an abelian category.If A is a commutative graded algebra finitely generated in degree 1 over k, then tails A is equivalent to the category coh(Proj A) of coherent sheaves on the projective scheme Proj A by Serre's theorem [18].For this reason, tails A is called the noncommutative projective scheme associated to A (see [1] for details).
The quotient functor is denoted by π : grmod A → tails A. We usually denote by M = πM ∈ tails A the image of M ∈ grmod A. Note that the k-linear autoequivalence M → M(n) preserves torsion modules, so it induces a k-linear autoequivalence M → M(n) for tails A, again called the shift functor.For M, N ∈ tails A, we write the vector space Ext i A (M, N ) := Ext i tails A (M, N ) and the graded vector space as before.
For an abelian category C , we define the global dimension of C by gldim C := sup{i | Ext i C (M, N ) = 0 for some M, N ∈ C }.The notion of graded isolated singularity for a noncommutative connected graded algebra A has been defined using the noncommutative projective scheme tails A (see [20], [10]).Definition 1.6.A graded right coherent connected graded algebra A is called a graded isolated singularity if gldim(tails A) < ∞.

Preliminaries
2.1.Ampleness.The ampleness of a line bundle is essential to construct a homogeneous coordinate ring of a projective scheme in commutative algebraic geometry.We will define a notion of ampleness in noncommutative algebraic geometry.Definition 2.1.
(1) An algebraic triple consists of a k-linear category C , an object O ∈ C , and a k-linear autoequivalence s ∈ Aut k C .In this case, we also say that (O, s) is an algebraic pair for C .
(2) A morphism of algebraic triples (F, θ, µ) (5) For an algebraic triple (C , O, s), we define a graded algebra by where the multiplication is given by the following rule: For an object M in C , we define a graded right B(C , O, s)-module where the right action is given by the following rule: Example 2.2.For an algebraic triple (C , O, s) and r ∈ N + , if A = B(C , O, s), then A (r) ∼ = B(C , O, s r ) and A [r] ∼ = B(C , r−1 i=0 s i O, s r ) (see [13]).Remark 2.3.A morphism of algebraic triples (F, θ, µ) Example 2.4.If A is a graded right coherent algebra, then π : grmod A → tails A induces a morphism of algebraic triples (grmod A, A, (1)) → (tails A, A, (1)), which induces a graded algebra homomorphism Moreover, for M ∈ grmod A, we have a graded right A-module homomorphism where we view H 0 (M) as a graded right A-module via φ A .
The following notion of ampleness introduced in [1] is a key concept in noncommutative projective geometry.Definition 2.5.We say that an algebraic pair (O, s) for a k-linear abelian category C is ample if (A1) for every M ∈ C , there exists an epimorphism p j=1 s −i j O → M in C for some i 1 , . . ., i p ≥ 0, and (A2) for every epimorphism φ : M → N in C , there exists m ∈ Z such that Theorem 2.6.The following statements hold.
(1) Let A be a graded algebra.If Proof.
(1) First we check that tails A is Hom-finite.It is enough to show that Hom A (A, M) = H 0 (M) 0 is finite dimensional over k for any M ∈ grmod A. The condition (b) says that H 0 (M) ≥0 is graded right coherent, so we have a surjection F → H 0 (M) ≥0 in grmod A where F is a finitely generated graded free right A-module.Since A is locally finite, we have dim k H 0 (M) 0 < ∞.
To prove that (A, (1)) satisfies (A1), it is enough to check that there exist positive integers i 1 , . . ., i p ∈ N + and an epimorphism p j=1 A(−i j ) → A. Since A is graded right coherent, we have an exact sequence this induces a desired epimorphism, so (A1) follows.
We next show that (A, (1)) satisfies (A2).First, note that, for every M ∈ grmod A and every n ∈ Z, since A is graded right coherent and M/M ≥n is finite dimensional, M/M ≥n is graded right coherent by Proposition 1.5, so M ≥n is also graded right coherent.Let φ : M → N be an epimorphism in tails A. Then there exists a homomorphism ψ : / / N ≥n are surjective for all n ≫ 0. Since πM ≥n ∼ = πM, πN ≥n ∼ = πN, we may assume that there exists an epimorphism ψ : is surjective for every i ≥ m.
Definition 2.7.Let A be a graded algebra.A twisting system on A is a sequence θ = {θ i } i∈Z of graded k-linear automorphisms of A such that θ i (xθ j (y)) = θ i (x)θ i+j (y) for every i, j ∈ Z and every x ∈ A j , y ∈ A. The twisted graded algebra of A by a twisting system θ is a graded algebra A θ where A θ = A as a graded k-vector space with the new multiplication x * y = xθ j (y) for x ∈ A j , y ∈ A.
If σ ∈ GrAut A is a graded algebra automorphism of A, then {σ i } i∈Z is a twisting system of A. In this case, we simply write A σ := A {σ i } .If B is a twisted graded algebra of A by a twisting system, then GrMod A ∼ = GrMod B by [25].
There is another notion of ampleness introduced in [11].For a ring R, a two-sided tilting complex Definition 2.9.Let R be a finite dimensional algebra and L a two-sided tilting complex of R.
(1) We say that L is quasi-ample if h q (L ⊗ L R i ) = 0 for all q = 0 and all i ≥ 0. (2) We say that L is ample if L is quasi-ample and (D L,≥0 , D L,≤0 ) is a t-structure on D b (mod R) where The heart of this t-structure is denoted by 10.The notions of ample and Fano in the above definition were called extremely ample and extremely Fano in [11], [12], [13].(

AS-regular
For an AS-regular algebra A over R of Gorenstein parameter ℓ, we define the Beilinson algebra of A by By [12, Corollary 3.7], a usual AS-regular algebra defined in Definition 1.3 is exactly an AS-regular algebra over k in the above definition.A typical example of an AS-regular algebra over R is given as follows.For a quasi-Fano algebra R of global dimension n, the preprojective algebra of R is defined as the tensor algebra ΠR := T R (Ext n R (DR, R)).
Conversely, if A is a graded right coherent AS-regular algebra over R of dimension d ≥ 1, then ∇A is a Fano algebra of gldim ∇A = d − 1 and grmod Π∇A ∼ = grmod A. Definition 2.13 ([12, Definition 3.9]).A locally finite N-graded algebra A with A 0 = R is called ASF-regular of dimension d and of Gorenstein parameter ℓ if the following conditions are satisfied: ( Remark 2.14.In the definition of an ASF-regular algebra given in [12, Definition 3.9], the condition gldim R < ∞ was not imposed.In this paper, we impose this condition to show that AS-regularity over R and ASF-regularity are equivalent.
In [12], Minamoto and the first author showed the following.
Theorem 2.15 ([12, Theorem 3.12.]).If A is an AS-regular algebra over R of dimension d and of Gorenstein parameter ℓ, then A is an ASF-regular algebra of dimension d and of Gorenstein parameter ℓ.
It was proved that the converse of Theorem 2.15 is also true when A is noetherian (see [22,Theorem 2.10]).For the purpose of this paper, we here show that a non-noetherian version of the converse of Theorem 2.15.Definition 2.16.For a locally finite N-graded algebra A, we say that the condition (EF) holds if every finite dimensional graded right A-module is graded right coherent.
If A is graded right coherent (in particular, right noetherian), then A satisfies (EF) by Proposition 1.5.If A is connected graded, then (EF) is equivalent to Ext-finiteness (that is, Ext i A (k, k) is finite dimensional for every i).Lemma 2.17.Let A be a locally finite N-graded algebra satisfying (EF).Then RΓ m (−) commutes with direct limits.
Proof.The proof is similar to that of [23,Lemma 4.3] by using (EF) instead of Extfiniteness.
Let A, C be graded algebras.Note that if M is a complex of graded C-A bimodules, then DM defined by (DM) i = D(M −i ) is a complex of graded A-C bimodules.

Theorem 2.18 (Local Duality).
Let A be a locally finite N-graded algebra, and C another N-graded algebra.Assume that A has finite cohomological dimension, and it satisfies (EF).Then for any Proof.Using Lemma 2.17, the proof works along the same lines as that of [23,Theorem 5.1].
If A is an ASF-regular algebra, then there exists a graded algebra automorphism ν of A such that D RΓ m (A) ∼ = A ν (−ℓ) [d] in D(GrMod A e ), 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 ω A := A ν (−ℓ) the canonical module over A (see [12,Section 3.2]).
Hence the result follows.
Remark 2.20.Let A be a graded right coherent algebra.Since A satisfies (EF) by Proposition 1.5, A is an ASF-regular algebra of dimension d and of Gorenstein parameter ℓ if and only if A is an AS-regular algebra over R = A 0 of dimension d and of Gorenstein parameter ℓ.Note that it is conjectured that every AS-regular algebra is graded right coherent.

Regular Tilting Objects and Relative Helices
3.1.Canonical Bimodules.The canonical sheaf plays an essential role to study a projective scheme in commutative algebraic geometry.We will define a notion of canonical bimodule for an abelian category.(1) If X is a smooth projective scheme, then the canonical sheaf ω X over X is the canonical bimodule for coh X.
(2) Let A be a noetherian AS-Gorenstein algebra over k, and ω A the canonical module of A. Then A is a graded isolated singularity if and only if ω A := πω A is the canonical bimodule for tails A ([20, Theorem 1.3]).(3) If A is a graded right coherent AS-regular algebra over R, then ω A := πω A is the canonical bimodule for tails A where ω A is the canonical module over A ([12, Theorem 4.12]).[11,Corollary 3.6, Corollary 3.12]), so ω R is a canonical bimodule for H ω −1 R .

Regular Tilting Objects.
Let T be a triangulated category.For a set of objects {E 0 , . . ., E r−1 } in T , we denote by E 0 , . . ., E r−1 the smallest full triangulated subcategory of T containing E 0 , . . ., E r−1 closed under isomorphisms and direct summands.
The following theorem can be derived from [6, Theorem 7].For the convenience of the reader, we include our own proof.
This induces the following commutative diagram where for all q = 0 and all i ≥ 0, we see that L is a quasi-ample two-sided tilting complex of R.
by Remark 3.8, Remark 3.5 (3), and the uniqueness of the Serre functor.Since ω −1 R is quasi-ample, we have = 0 for all q = 0 and all i ≥ 0, so T is a regular tilting object of D b (C ).Proof.Since T is regular tilting for D b (C ), R is a quasi-Fano algebra of gldim R = gldim C by Theorem 3.10, so ω −1 R is a quasi-ample two-sided tilting complex of R. The commutative diagram (3.1) induces the following isomorphisms of graded algebras R is an ample two-sided tilting complex of R by [11, Theorem 3.7], hence R is a Fano algebra of gldim R = gldim C .By Theorem 2.12, is a graded right coherent AS-regular (Calabi-Yau) algebra over R of dimension gldim R + 1 = gldim C + 1 and of Gorenstein parameter 1.

Relative Helices.
In this subsection, we will define a "relaxed" version of a helix.Definition 3.12.Let T be a k-linear triangulated category.
A (relative) exceptional sequence {E, F } consisting of two objects is called a (relative) exceptional pair.
(1) A sequence of objects = 0 for every q = 0 and every i ≤ j.
Definition 3.15.Let T be a k-linear triangulated category.For a pair of objects {E, F } in T , we define Hom with trivial differentials.Moreover we define objects L E F and R F E in T by using distinguished triangles It is known that if {E, F } is an exceptional pair, then {L E F, E} and {F, R F E} are both exceptional pairs, and Mutations of exceptional pairs can be extended to mutations of exceptional sequences.For a sequence of objects ǫ = {E 0 , . . ., E ℓ−1 }, we define be a sequence of objects in a k-linear triangulated category.For each i = 0, . . ., ℓ − 2, the following are equivalent: (1) ǫ is a (full) exceptional sequence.
We inductively define Remark 3.17.Let C be a k-linear abelian category.There is another definition of a helix, which requires the condition (H2)' in place of (H2) (see [13,Definition 4.3]).If C has the canonical bimodule ω C , then [3,Assertion 4.2] and Remark 3.5 (3), so the above definition of a helix agrees with the one given in [13,Definition 4.3] if and only if ℓ = gldim C + 1. Lemma 3.18.Let C be a k-linear abelian category having the canonical bimodule ω C , and {E i } i∈Z a (full) geometric relative helix of period ℓ for D b (C ).For r ∈ N + such that r | ℓ, { i∈I j E i } j∈Z where I j = {i ∈ Z | jr ≤ i ≤ (j +1)r −1} is a (full) geometric relative helix of period ℓ/r for D b (C ).In particular, for an algebraic pair (O, s) for C , if {s i O} i∈Z is a (full) geometric relative helix of period ℓ for D b (C ), then {s jr ( r−1 i=0 s i O)} j∈Z is a (full) geometric relative helix of period ℓ/r for D b (C ).
Proof.First, we show that (H1), that is, { i∈I j E i , . . ., i∈I j+ℓ/r−1 E i } is a relative exceptional sequence for D b (C ) for every j ∈ Z.
(RE2) Using the facts that {E jr , . . ., E (j+1)r−1 } is a relative exceptional sequence for any j ∈ Z and {E i } i∈Z is geometric, we have Hom C ( i∈I j E i , i∈I j E i [q]) = 0 for every q = 0 and every j = 0, . . ., ℓ/r − 1.
(RE3) For any i ∈ Z and any i ≤

Proof. By definition, gldim End
) = 0 for all q = 0 and all j ≥ 0, so E i is regular tilting.

Lemma 3.20. Let C be a k-linear abelian category having the canonical bimodule ω
For every q = 0 and every i ≤ j, Clearly, (A1) for the pair (O, s) is equivalent to (A1) for the pair ( i∈I 0 s i O, s r ).
For every epimorphism φ : M → N in C , we see that ) is surjective for every i ∈ I j .This implies that (A2) for (O, s) is equivalent to (A2) for ( i∈I 0 s i O, s r ).(1) Proof.(1) If {s i O} i∈Z is a full geometric relative helix of period ℓ, then {s jℓ ( ℓ−1 i=0 s i O)} j∈Z is a full geometric relative helix of period 1 by Lemma 3.18, so T := ℓ−1 i=0 s i O ∈ C is a regular tilting object for D b (C ) by Lemma 3.19.

Main Result
We are now ready to state and prove the main result of this paper, which gives a complete answer to Question 1.1.Note that if A is an AS-regular algebra over A 0 of dimension 0, then A is finite dimensional over k, so tails A is trivial.
We next show that {A(i)} i∈Z is a full geometric relative helix of period ℓ for D b (tails A).By [12,Proposition 4.4 [12,Proposition 4.4] again that {A(i)} i∈Z is a geometric relative helix of period ℓ.Furthermore, similar to the proof of [12, Proposition 4.3], we have A(i), . . ., A(i + ℓ − 1) = D b (tails A) for every i ∈ Z, so {A(i)} i∈Z is a full relative helix.

], End
Since C ∼ = tails A, we see that C has an ample algebraic pair (O, s) such that {s i O} i∈Z is a full geometric relative helix of period ℓ for D b (C ).
For the rest, we will show that A is AS-regular over A 0 = End C (O) of dimension n + 1 and of Gorenstein parameter ℓ.
First assume n = 0.It follows from Theorem 3.11 that gldim R = gldim where deg x = 1 by Lemma 2.8 (1).Since {O, . . ., s ℓ−1 O} is a relative exceptional sequence, Hom C (O, where A 0 is semisimple and deg x = ℓ, so A is AS-regular over A 0 of dimension 1 and of Gorenstein parameter ℓ. We now assume n ≥ 1.Since A is graded right coherent, it satisfies the condition (EF), so it is enough to show that A is ASF-regular of dimension n + 1 and of Gorenstein parameter ℓ by Theorem 2.19.Note that we have an exact sequence A) → 0 and isomorphisms H q m (A) ∼ = H q−1 (A), q ≥ 2 of graded A-A bimodules where φ A : A → H 0 (A) is the graded algebra homomorphism defined in Example 2.4.Thus it is enough to check that φ A above is an isomorphism and H q (A) ∼ = 0 if q = 0, n DA(ℓ) if q = n as graded right and left A-modules.
If j ≥ 0, then Ext q C (O, s j O) = 0 for all q = 0 since {s j O} is geometric.
= 0 for all q = n since {s j O} is geometric.It follows that H q (A) ∼ = j∈Z Ext q C (O, s j O) = 0 for all q = 0, n.On the other hand, if −ℓ < j < 0, then Hom C (O, Recall that the functor π : grmod A → tails A induces a morphism of algebraic triples (grmod A, A, (1)) → (tails A, A, ψ is an isomorphism of a graded algebras by Theorem 2.6 (2), so φ A is also an isomorphism of graded algebras.

Consider the diagram H
where the top and the bottom squares are commutative and F is a map induced by the Serre functor as in Remark 3.2.For (α, In this case, A is Koszul if and only if ℓ = gldim C + 1 (cf.Remark 3.17).
Proof.Note that if A is AS-regular over k of dimension d and of Gorenstein parameter ℓ, and  Example 4.5.In the above Corollary, the condition r | ℓ cannot be dropped.For example, if A = k[x] with deg x = 3, then A is AS-regular over k of dimension 1 and of Gorenstein parameter 3.If B := A [2] is AS-regular over B 0 , then gldim B = gldim A = 1, so B = B 0 [x] as a graded vector space by the proof of Theorem 4.1 (see also [12,Theorem 4.15]).Since it is not the case, so B is not AS-regular over B 0 = k × k.Since GrMod A [r] ∼ = GrMod A for every r ∈ N + , this example shows that AS-regularity is not a graded Morita invariant if we do not require algebras to be connected graded (compare with [25, Theorem 1.3]).
5. Smooth Quadric Surfaces in a Quantum P 3 It is well-known that, for a smooth quadric surface Q in P 3 , there exists a noetherian AS-regular algebra B = k x, y /(x 2 y − yx 2 , xy 2 − y 2 x) of dimension 3 and of Gorenstein parameter 4 such that coh Q ∼ = tails B. In this section, we will prove a noncommutative generalization of this result as an application of the main result of this paper.
Throughout this section, we assume that k is an algebraically closed field of characteristic 0. Definition 5.1 ([19]).We say that a k-linear abelian category C is a smooth quadric surface in a quantum P 3 if C ∼ = tails S/(f ) where S is a 4-dimensional noetherian quadratic AS-regular algebra over k and f ∈ S 2 is a central regular element such that A S/(f ) is a domain and a graded isolated singularity.
Let S be a 4-dimensional noetherian quadratic AS-regular algebra over k.Then the Hilbert series of S is H S (t) = 1/(1 − t) 4 , and S is a Koszul domain.Let f ∈ S 2 be a central regular element and A = S/(f ).Then A is a noetherian AS-Gorenstein Koszul algebra of dimension 3 and of Gorenstein parameter 2. There exists a central regular element z ∈ A ! of degree 2 such that A ! /(z) ∼ = S !where A ! , S ! are Koszul duals of A, S. We define C(A) := A ! [z −1 ] 0 .
We call M ∈ grmod A graded maximal Cohen-Macaulay if depth M = lcd M = lcd A (= 3) or M = 0.It is well-known that M ∈ grmod A is graded maximal Cohen-Macaulay if and only if Ext i A (M, A) = 0 for all i = 0. We write CM Z (A) for the full subcategory of grmod A consisting of graded maximal Cohen-Macaulay modules.Proposition 5.2.Let S be a 4-dimensional noetherian quadratic AS-regular algebra over k, and f ∈ S 2 a central regular element.If A = S/(f ) is a domain, then the following are equivalent: (1) A is a graded isolated singularity.
(2) A is of finite Cohen-Macaulay representation type (i.e, there exist only finitely many indecomposable graded maximal Cohen-Macaulay modules up to isomorphisms and degree shifts).For the rest, we assume that S is a 4-dimensional noetherian quadratic AS-regular algebra over k, f ∈ S 2 is a central regular element, and A = S/(f ) is a domain and a graded isolated singularity.In this case, gldim(tails A) = 2 and tails A has the canonical bimodule ω A such that M ⊗ A ω A = M ν (−2) for M ∈ tails A by Example 3.6 where ν is The Auslander-Reiten quiver of CM Z (A) is given as follows: where dotted arrows show the Auslander-Reiten translation τ in CM Z (A).(There are no arrows.) Lemma 5.6.The following hold.
By the structure of the Auslander-Reiten quiver of CM Z (A), this is zero for any i.
For an abelian category C , we denote by D(C ) the derived category of C and by D b (C ) the bounded derived category of C .For M, N ∈ D(C ), we often write Hom C (M, N ) := Hom D(C ) (M, N ) by abuse of notation.For M, N ∈ D(C ) and i ∈ Z, we set Ext (a) A is graded right coherent, and (b) for any M ∈ tails A and any n ∈ Z, H 0 (M) ≥n is graded right coherent, then tails A is Hom-finite k-linear abelian category and (A, (1)) is an ample pair for tails A. (2) Conversely, if (O, s) is an ample pair for a Hom-finite k-linear abelian category C , then (a) A := B(C , O, s) ≥0 is a graded right coherent algebra, (b) for any M ∈ C and any n ∈ Z, H 0 (M) ≥n is graded right coherent, and (c) the functor C → tails A; M → π H 0 (M) ≥0 induces an isomorphism of algebraic triples (C , O, s) ∼ = (tails A, A, (1)).

Lemma 2 . 8 .
Let C and C ′ be k-linear abelian categories.If (C , O, s) and (C ′ , O ′ , s ′ ) are equivalent algebraic triples, then the following hold.

(
Algebras over R. Two generalizations of a notion of AS-regularity were introduced in [12].Definition 2.11 ([12, Definition 3.1]).A locally finite N-graded algebra A with A 0 = R is called AS-regular over R of dimension d and of Gorenstein parameter ℓ if the following conditions are satisfied:

Theorem 2 . 19 .
If A is an ASF-regular algebra of dimension d and of Gorenstein parameter ℓ satisfying (EF), then A is an AS-regular algebra over R = A 0 of dimension d and of Gorenstein parameter ℓ.Proof.Since A is ASF-regular, we have D RΓ m (A) ∼ = A ν (−ℓ)[d] in D(GrMod A e ).It follows from Theorem 2.18 that

Definition 3 . 1 ([ 4 ,
Definition 3.1]).Let C be a Hom-finite k-linear category.A Serre functor for C is a k-linear autoequivalence S ∈ Aut k C such that there exists a bifunctorial isomorphism F X,Y : Hom C (X, Y ) → D Hom C (Y, S(X)) for X, Y ∈ C .Remark 3.2.We explain the functoriality of a Serre functor S in X in the above definition.Define functors G = Hom C (−, Y ) and H = D Hom C

Definition 3 . 3 .
Let C be an abelian category.A bimodule M over C is an adjoint pair of functors from C to itself with the suggestive notationM = (− ⊗ C M, Hom C (M, −)).A bimodule M over C is invertible if − ⊗ C M is an autoequivalence of C .In this case, the inverse bimodule of M is defined by M −1 = (− ⊗ C M −1 , Hom C (M −1 , −)) := (Hom C (M, −), − ⊗ C M).Definition 3.4.Let C be a k-linear abelian category.A canonical bimodule for C is an invertible bimodule ω C over C such that, for some n ∈ Z, the autoequivalence − ⊗ L C ω C [n] of D b (C ) induced by − ⊗ C ω C is a Serre functor for D b (C ).Remark 3.5.Let C be a k-linear abelian category.(1) Since the Serre functor for D b (C ) is unique, a canonical bimodule for C is unique if it exists.(2) If C has a canonical bimodule, then D b (C ) has a Serre functor by definition, so D b (C ) is automatically Hom-finite.(3) If C has a canonical bimodule ω C , and − ⊗ L C ω C [n] is the Serre functor for D b (C ), then it is easy to see that gldim C = n < ∞.Example 3.6.

Definition 3 . 7 .Definition 3 . 9 .
Let T be a triangulated category.An object T ∈ T is called tilting if (1) T = T , and (2) Hom T (T, T [q]) = 0 for all q = 0. Remark 3.8.If C is a k-linear abelian category such that D b (C ) is Hom-finite, then it is known that D b (C ) is an algebraic triangulated category (see [7, Section 1.2 and Section 3.1]) and Krull-Schmidt (see [8, Corollary A.2] and [2, Corollary 2.10]).Hence, if T is a tilting object for D b (C ) such that gldim End C (T ) < ∞, then the functor RHom C (T, −) : D b (C ) → D b (mod End C (T )) gives an equivalence of triangulated categories by [9, Theorem 2.2].Let C be a k-linear abelian category having the canonical bimodule ω C .We say that an object T ∈ D b (C ) is regular tilting if (RT1) gldim End C (T ) < ∞, (RT2) D b (C ) = T , and (RT3) Hom C (T, T ⊗

Theorem 3 . 10 .
Let C be a k-linear abelian category with the canonical bimodule ω C , and T ∈ D b (C ) a tilting object.Then T is regular tilting if and only if R := End C (T ) is a quasi-Fano algebra of gldim R = gldim C .Proof.Note that since C is assumed to have a canonical bimodule, D b (C ) is Hom-finite.(⇒) Assume that T is a regular tilting object of D b (C ).Let − ⊗ L C ω C [m] be the Serre functor for D b (C ) and let gldim R = n.Then we have m = gldim C by Remark 3.5 (3).Using Remark 3.8 and the uniqueness of the Serre functor, we have the following commutative diagram algebra of gldim R = m = gldim C by [12, Remark 1.3] (cf.[11, Remark 4.4]).(⇐) If T ∈ C is a tilting object for D b (C ) and R = End C (T ) is a quasi-Fano algebra of gldim R = gldim C , then we have the following commutative diagram
H2) is satisfied, so { i∈I j E i } j∈Z is a relative helix of period ℓ/r for D b (C ).The full and geometric properties are straightforward.Lemma 3.19.Let C be a k-linear abelian category having the canonical bimodule ω C .If {E i } i∈Z is a full geometric relative helix of period 1 for D b (C ), then E i is a regular tilting object of D b (C ) for every i ∈ Z.

Proposition 3 . 22 .
Let C be a k-linear abelian category having the canonical bimodule ω C , and (O, s) an algebraic pair for C .If {s i O} i∈Z is a full geometric relative helix of period ℓ for D b (C ), then the following hold.

Theorem 4 . 1 .
Let C be a k-linear abelian category.Then C ∼ = tails A for some graded right coherent AS-regular algebra over A 0 of dimension at least 1 and of Gorenstein parameter ℓ if and only if (AS1) C has a canonical bimodule ω C , and (AS2) there exists an ample algebraic pair (O, s) for C such that {s i O} i∈Z is a full geometric relative helix of period ℓ for D b (C ).In fact, if (AS1) and (AS2) are satisfied, then A = B(C , O, s) ≥0 is a graded right coherent AS-regular algebra over A 0 = End C (O) of dimension gldim C + 1 and of Gorenstein parameter ℓ such that C ∼ = tails A. In this case, A is right noetherian if and only if O ∈ C is a noetherian object.Proof.(⇒) Let A be a graded right coherent AS-regular algebra over A 0 of dimension d ≥ 1 and of Gorenstein parameter ℓ.Then tails A has the canonical bimodule ω A by Example 3.6.By Theorem 2.12, R := ∇A is a Fano algebra of gldim R = d − 1, and B := ΠR is a graded right coherent AS-regular algebra over R of dimension d and of Gorenstein parameter 1.Since B is a twisted graded algebra of A [ℓ] by a graded algebra automorphism by [12, Theorem 4.12], there exists an equivalence functor grmod A → grmod B sending ℓ−1 i=0 A(i) to B by [12, Remark 4.9].Since we have A (⇐) Suppose that C satisfies (AS1) and (AS2).Let n = gldim C .Since (O, s) is ample for C , A := B(C , O, s) ≥0 is graded right coherent and (C , O, s) ∼ = (tails A, A, (1)) by Theorem 2.6.By Proposition 3.22, T := ℓ−1 i=0 s i O ∈ C is a regular tilting object for D b (C ) and (T, − ⊗ C ω −1 C ) is ample for C , so it follows from Theorem 3.11 that ΠR ∼ = B(C , T, − ⊗ C ω −1 C ) ≥0 is a graded right coherent AS-regular algebra over R := End C (T ) of dimension n+1.Moreover, since {s −ℓ+1 O, . . ., s −1 O, O} is a relative exceptional sequence, (1)) by Example 2.4.Since C is Hom-finite and (O, s) is ample for C , we have a functor H 0 (−) ≥0 : C → grmod A by Theorem 2.6(2).Since H0 (−) ≥0 • s = ∞ i=0 Hom C (O, s i+1 (−)) and (1) • H 0 (−) ≥0 = ∞ i=−1 Hom C (O, s i+1 (−)), there exists a natural transformation H 0 (−) ≥0 • s → (1) • H 0 (−) ≥0 .Since H 0 (O) ≥0 = B(C , O, s) ≥0 = A,the functor H 0 (−) ≥0 : C → grmod A induces a morphism of algebraic triples (C , O, s) → (grmod A, A, (1)).By Theorem 2.6 (2), the composition of these morphisms is an isomorphism of algebraic triples (C , O, s) → (tails A, A, (1)).In the commutative diagram ψ : B(C , O, s) H 0 (−) ≥0 − −−−− → B(grmod A, A, (1)) π − −− → B(tails A, A, (1)) for every γ ∈ Hom C (s i+j+ℓ O, O) by Remark 3.2, the above diagram commutes, so H n (A) ∼ = DA(ℓ) as graded right A-modules.Similarly, we can show that H n (A) ∼ = DA(ℓ) as graded left A-modules.Hence A is ASF-regular of dimension n + 1 ≥ 2 and of Gorenstein parameter ℓ.For the last statement, since C is Hom-finite, H 0 (M) is finite dimensional for ev-ery object M ∈ C .Since (O, s) is ample for C , if O ∈ C is a noetherian object, then A = B(tails A, O, s) ≥0 is right noetherian by [1, Theorem 4.5].Conversely, since (C , O, s) ∼ = (tails A, A, (1)), if A = B(tails A, O, s) ≥0 is right noetherian, then A ∈ tails A is a noetherian object, so O ∈ C is a noetherian object.Corollary 4.2.Let C be a k-linear abelian category.Then C ∼ = tails A for some graded right coherent AS-regular algebra over k of dimension at least 1 and of Gorenstein parameter ℓ if and only if (AS1) C has a canonical bimodule ω C , and (AS2) there exists an ample algebraic pair (O, s) such that {s i O} i∈Z is a full geometric helix of period ℓ for D b (C ).In fact, if (AS1) and (AS2) are satisfied, then A = B(C , O, s) ≥0 is a graded right coherent AS-regular algebra over k of dimension gldim C + 1 and of Gorenstein parameter ℓ such that C ∼ = tails A.

Corollary 4 . 3 .
the minimal free resolution of k over A, then F d ∼ = A(−ℓ), so k has a linear resolution if and only if ℓ = d.In the above setting, d = gldim C + 1, so the last statement holds.Let C be a k-linear abelian category.Then C ∼ = tails A for some graded right coherent AS-regular algebra over A 0 of dimension at least 1 if and only if (AS1) C has a canonical bimodule ω C , and (AS2)' there exists a regular tilting object T ∈ C for D b (C ) such that (T, − ⊗ C ω −1 C ) is ample for C .Proof.If C ∼ = tails A for some graded right coherent AS-regular algebra over A 0 of Gorenstein parameter ℓ, then C has a canonical bimodule ω C , and there exists an ample algebraic pair (O, s) for C such that {s i O} i∈Z is a full geometric relative helix of period ℓ for D b (C ) by Theorem 4.1.By Proposition 3.22, T : a full geometric relative helix of period 1 for D b (C ) by Lemma 3.20, so the result follows from Theorem 4.1.Corollary 4.4.Let A be a graded right coherent (noetherian) AS-regular algebra over A 0 of dimension d and of Gorenstein parameter ℓ.For r ∈ N + such that r | ℓ, B := A [r] is a graded right coherent (noetherian) AS-regular algebra over B 0 of dimension d and of Gorenstein parameter ℓ/r.Proof.Since (A, (1)) is an ample algebraic pair for tails A such that {A(i)} i∈Z is a full geometric relative helix of period ℓ for D b (tails A) by the proof of Theorem 4.1, ( r−1 i=0 A(i), (r)) is an ample algebraic pair for tails A such that {( r−1 i=0 A(i))(rj)} j∈Z is a full geometric relative helix of period ℓ/r for D b (tails A) by Lemma 3.21 and Lemma 3.18.Since B = A [r] ∼ = B(D b (tails A), r−1 i=0 A(i), (r)) ≥0 , we see that B is a graded right coherent AS-regular algebra over B 0 of dimension gldim(tails A)+1 = d and of Gorenstein parameter ℓ/r by Theorem 4.1.
) Two algebraic triples (C , O, s) and (C ′ , O ′ , s ′ ) are isomorphic, denoted by (C , O, s) ∼ = (C ′ , O ′ , s ′ ) if there exists a morphism of algebraic triples (F, θ, µ) : (C , O, s) → (C ′ , O ′ , s ′ ) such that F is an equivalence functor and µ is a natural isomorphism.(4) Two algebraic triples (C , O, s) and (C ′ , O ′ , s ′ ) are equivalent, denoted by (C , O, s) ∼ (C ′ , O ′ , s ′ ) if there exists an equivalence functor modules (see Example 2.4).Since the two middle terms in the above sequence are graded right coherent, we see that H 0 m (M) ≥n and H 1 m (M) ≥n are graded right coherent.Moreover, since H 0 m (M) and H 1 m (M) are m-torsion modules, so are H 0 m (M) ≥n and H 1 m (M) ≥n .These imply that H 0 m (M) ≥n and H 1 m (M) ≥n are finite dimensional over k.Hence (φ M ) ≥n : M ≥n → H 0 (M) ≥n is an isomorphism in grmod A for every n ≫ 0. By applying the same argument for N, there exists m Theorem 2.12 ([11, Corollary 3.12], [12, Theorem 4.2, Theorem 4.12, Theoerm 4.14]).If R is a Fano algebra, then ΠR