On Ideal Classes of Three Dimensional Sklyanin Algebras

On ideal classes of three dimensional Sklyanin algebras

4 exposésau seminaire d’algèbre UniversitéJean Monnet, Saint-Etienne, France by Koen De Naeghel University of Hasselt, Belgium May - June 2005

Abstract These notes reﬂect a series of four lectures that I have given at the university of Jean Monnet in Saint-Etienne, France. The purpose of these talks was to outline the paper [13] in more detail than a regular talk. Of course it was not my intention to prove every theorem in full detail. Rather, the idea was to show the path towards the results. This means I had to make a selection which proofs to discuss - and which not. I hope that the reader agrees in my selection. I also wanted to stress the computational methods on Hilbert series, Euler forms and Grothendieck groups. I am greatful for the pre-doctoral position oﬀered by the European research training network Liegrits.

I would like to thank Roland Berger and Nicolas Marconnet for their kind inviation and the joint research we have done in Saint-Etienne.

Also, I want to express my sincere thanks to my thesis advisor Michel Van den Bergh for introducing me to the subject and for sharing with me his mathematical ideas. Contents

1 Introduction and motivation 3 1.1 Hilbert schemes on aﬃne planes ...... 3 1.1.1 The commutative polynomial algebra k[x, y] ...... 3 1.1.2 The ﬁrst Weyl algebra ...... 4 1.2 Hilbert scheme of projective planes ...... 6 1.2.1 The commutative polynomial algebra k[x, y, z] ...... 6 1.2.2 The homogenized Weyl algebra ...... 8 1.2.3 Three dimensional Sklyanin algebras ...... 10

2 Preliminaries and basic tools 12 2.1 Connected graded algebras and Hilbert series ...... 12 2.2 Three dimensional Artin-Schelter regular algebras ...... 13 2.3 Modules over quadratic Artin-Schelter regular algebras ...... 16 2.3.1 Hilbert series and gk-dimension ...... 17 2.3.2 Linear modules ...... 17 2.4 Quantum projective planes ...... 19 2.4.1 Projective schemes ...... 19 2.4.2 The Grothendieck group and the Euler form ...... 20 2.4.3 Serre duality ...... 21

3 From reﬂexive modules to the quantum plane 22 3.1 Reﬂexive modules and vector bundles ...... 22 2 3.2 The Grothendieck group and the Euler form for Pq ...... 23 3.3 Normalized rank one modules and sheaves ...... 24 3.4 Cohomology of line bundles on the quantum plane ...... 26

4 From the quantum plane to the elliptic curve 27 4.1 Geometric data associated to quantum planes ...... 27 4.2 Restriction of line bundles to the elliptic curve ...... 30

5 From the quantum plane to quiver representations 31 5.1 Generalized Beilinson equivalence ...... 31 5.2 The Grothendieck group and the Euler form for ∆ ...... 33 2 5.3 First description of Rn(Pq) ...... 33 5.4 Line bundles with invariant one ...... 36 5.5 Analogy with the homogenized Weyl algebra ...... 36 5.6 Induced Kronecker quiver representations ...... 38 5.7 Semistable representations ...... 39 2 5.8 Second description of Rn(Pq)...... 42

2 Throughout we ﬁx an algebraically closed ﬁeld k of characteristic 0.

1 Introduction and motivation 1.1 Hilbert schemes on aﬃne planes 1.1.1 The commutative polynomial algebra k[x, y]

Let A0 = k[x, y] denote the commutative polynomial algebra in two variables, which we view as the coordinate ring of the aﬃne plane A2. The Hilbert scheme of points on A2 parametizes the cyclic ﬁnite dimensional A0-modules

2 Hilbn(A ) = {V ∈ mod A0 | V cyclic and dimk V = n}/ iso (1)

2 For V ∈ Hilbn(A ) its annihilator AnnA0 (V ) = {a ∈ A0 | a · V = 0} is an ideal of A0 of ﬁnite codimension, and this correspondence is reversible:

2 Hilbn(A ) = {I ⊂ A0 ideal | dimk A0/I = n}/ iso

2 ` 2 Also, Hilb(A ) = Hilbn(A ) parameterizes the isomorphism classes of ﬁnitely generated torsion free rank one A0-modules:

2 a 2 Hilb(A ) = Hilbn(A ) (2) = R(A0) = { f.g. torsion free rank one A0-modules }/ iso since every such module is isomorphic to an unique ideal of ﬁnite codimension.

Finally, we may rephrase this into the language of quiver representations. 2 Let V ∈ Hilbn(A ) be a cyclic A0-module of dimension n. Multiplication by x and y on V induce linear maps on V represented by n × n matrices X, Y for which [X, Y] = 0. We also have a vector v ∈ V for which v · A0 = V . Thus 2 n [X, Y] = 0 V ∈ Hilbn(A ) 7→ data X, Y ∈ Mn(k), v ∈ k : n (3) khX, Yi · v = k Note that khX,Y i = k[X,Y ] since [X,Y ] = 0. Conversely, such data on the n right of (3) determine an A0-module structure on k which is cyclic, hence an 2 object in Hilbn(A ). Furthermore, isomorphism classes on the left are in one- to-one correspondence with the orbits of the group Gln(k) acting on the data on the right by (simultaneous) conjugation. Apparently, the conditions on the right of (3) may be replaced by - at ﬁrst sight weaker - conditions 2 n im([X, Y]) ⊂ k · v V ∈ Hilbn(A ) 7→ data X, Y ∈ Mn(k), v ∈ k : n (4) khX, Yi · v = k

3 Indeed, by standard arguments in linear algebra one shows that such data on the right of (4) imply [X, Y] = 0. See for example [21, §2.2]. Associated with data on the right of (4) are the linear maps i : k → kn : 1 7→ v n j : k → k : u 7→ j(u) such that [X, Y] · u = j(u) · v Now the quadruple (X, Y, i, j) may be visualized as

X i

j rr Y which determines a representation of the following quiver Q with dimension vector (n, 1)

Writing rep(n,1) Q for the representations of the quiver Q with dimension vector (n, 1) we ﬁnd

2 ∼ Hilbn(A ) = {(X, Y, i, j) ∈ rep(n,1) Q | [X, Y] = ij n and khX, Yi · i(1) = k }/ Gln(k) (5) where the group Gln(k) acts by conjugation −1 −1 −1 ∀g ∈ Gln(k):(X, Y, i, j) 7→ (gXg , gYg , gi, jg ) 2 2 2 Note again that in fact j = 0. Also, Hilb0(A ) is a point and Hilb1(A ) = A .

1.1.2 The ﬁrst Weyl algebra

Let A1 = khx, yi/(xy − yx − 1) be the first Weyl algebra. Thinking of A1 as a noncommutative version of A0 = k[x, y] we would like to have an analogue for the Hilbert scheme of points on A2. A first (naive) attempt based on (1) would be to consider cyclic finite dimensional right A1-modules

{V ∈ mod A1 | V cyclic and dimk V = n}

But in contrast with A0 this is the empty set for n > 0. Indeed, if there were such a module V then multiplication by x and y induce linear maps on V = kn

4 represented by n × n matrices X, Y. The relation xy − yx − 1 = 0 in A1 implies [Y, X] − I = 0. Taking the trace we obtain Tr(YX) − Tr(XY) − Tr(I) = Tr(0) i.e. n = 0. Similary,

{I ⊂ A1 right ideal | dimk A1/I = n} = ∅ for n > 0.

Thus there seems no reason to expect that there should be results for A1 similar to the ones indicated above for A0. But amazingly enough there are. The idea 2 is to consider the alternative description (2) of Hilbn(A ). Deﬁne

R(A1) = { ﬁnitely generated torsion free rank one A1-modules }/ iso Note that such modules are automatically reﬂexive. We recall the basic result.

Theorem 1.1. There exist smooth connected aﬃne varieties Cn of dimension 2n such that there is a natural bijection a Cn ↔ R(A1) = { f.g. torsion free rank one A1-modules }/ iso n where the variety Cn is the so-called nth Calogero-Moser space

Cn = {(X, Y, i, j) ∈ rep(n,1)(Q) | [X, Y] + I = ij}/ Gln(k) where Gln(k) acts by conjugation. Remark 1.2. 1. The first proof of Theorem 1.1 used the fact that there is a description of R(A1) in terms of the (infinite dimensional) adelic Grassma- nian, due to Cannings and Holland [12]. Using methods from integrable systems Wilson [31] established a relation between the adelic Grassmanian and the Calogero-Moser spaces. In fact, the orbits of the natural Aut(A1)- action on R(A1) are indexed by N, and the orbit corresponding to n is in natural bijection with the nth Calogero-Moser space Cn. The fact that ∼ R(A1)/ Aut(A1) = N has also been proved by Kouakou in his (unpub- lished) PhD thesis [16]. For more details on Calogero-Moser spaces, adelic Grassmanians, ideals of the first Weyl algebra and their interactions we also refer to [18].

2. At ﬁrst dight the description for Cn is not quite analogous as the commutative situation (5) since the stability condition is missing. But one may prove (see for example [18]) that the representations in Cn automatically satisfy khX,Y i · i(1) = kn. The fundamental reason for this is that the torsion free right ideals in A1 are automatically reﬂexive, while in the case of A0 they are not. 3. Note that we may simplify the description of the nth Caloger-Moser space Cn as 2 Cn = {(X, Y) ∈ Mn(k) | rank([Y, X] − I) ≤ 1}/ Gln(k)

where Gln(k) acts by simultaneous conjugation. From this description we 2 see that Cn ⊂ Mn(k) is a closed subvartiety of an aﬃne space, hence 2 aﬃne. Also note that C0 is a point and C1 = A .

5 1.2 Hilbert scheme of projective planes 1.2.1 The commutative polynomial algebra k[x, y, z] Let A = k[x, y, z] denote the commutative polynomial algebra in three variables, which we view as the coordinate ring of the projective plane P2. We 2 2 2 2 now consider the affine plane A as the open affine part A = P \ l∞ of P where the line l∞ given by the equation z = 0. There is a restriction functor i∗ : coh P2 → coh P1 associating with each sheaf its restriction to the line at infinity.

2 Let V ∈ Hilbn(A ) be a cyclic n-dimensional A0-module. Then V extends 2 to a zero dimensional subscheme X ∈ Hilbn(P ) with the property that multi- 0 2 ∼ 0 2 plication by z induces an isomorphisms H (P , OX (l − 1)) = H (P , OX (l)) for ∗ l 0. This means that i OX = 0. Writing IX for the ideal sheaf of OX we ∗ 1 have i IX = OP . These correspondences are reversible: 2 2 ∗ Hilbn(A ) = {X ∈ Hilbn(P ) | i OX = 0} 2 ∗ 1 = { torsion free rank one sheaves I ∈ coh P s.t. c2(I) = n, i I = OP }/ iso We will now recall how these objects may be described by their homology. We have an equivalence of derived categories, known as Beilinson equivalence [8]

RHom 2 (E,–) b 2 −→P b D (coh P ) ←− D (mod ∆) L – ⊗∆E

2 2 2 where E = OP ⊕ OP (−1) ⊕ OP (−2) and mod ∆ is the category of ﬁnite dimensional representations of the quiver ∆

0 −→u −→u 0 −→v −→v w w0 r −→ r −→ r with relations reﬂecting the relations in A = k[x, y, z]

 0 0  uv = vu vw0 = wv0  wu0 = u0w

2 Under this equivalence, an object X ∈ Hilbn(P ) is determined by a representation N of ∆

0 −→X −→X 0 0 2 Y 0 2 Y 0 2 H (P , OX ) −→ H (P , OX (1)) −→ H (P , OX (2)) 0 −→Z −→Z

6 where the linear map X are induced by multiplication by x, etc. and Y 0X = 0 X Y etc. (matrices will always be acting on the left). Shifting OX if necessary, this representation N has dimension vector (n, n, n). As pointed out above the linear maps Z and Z0 are isomorphisms. By an argument of Baer [7], N is actually determined by the linear maps X,Y,Z on the left, which is a representation of the Kronecker quiver ∆0

−→u −→v w r −→ r Furthermore, consideration of matrix multiplications  −YZ 0  0 0 0 X Y Z ·  X 0 Z  = 0 0 −X −Y | {z } A(X,Y,Z) and the fact that Z0 is an isomorphism yields ker A(X,Y,Z) ≥ n hence 2 rank A(X,Y,Z) ≤ 2n. This leads to a description of Hilbn(A ) in terms of quiver representations of the Kronecker quiver ∆0:

2 0 Hilbn(A ) = {(X,Y,Z) ∈ rep(n,n) ∆ | Z isomorphism , rank A(X,Y,Z) ≤ 2n, −1 −1 n n khZ X,Z Y i · v = k for some v ∈ k }/ Gln(k)

Indeed: Putting X = Z−1X, Y = Z−1Y we ﬁnd the earlier description of 2 Hilbn(A ): 0 = {(X, Y, I) ∈ rep(n,n) ∆ | rank A(X, Y, I) ≤ 2n, n n khX, Yi · v = k for some v ∈ k }/ Gln(k) 0 = {(X, Y, I) ∈ rep(n,n) ∆ | [X, Y] = 0, n n khX, Yi · v = k for some v ∈ k }/ Gln(k) where one uses  −1      Z 0 0 −YZ 0 −YI 0  0 Z−1 0  ·  X 0 Z  =  X 0 I  0 0 Z−1 0 −X −Y 0 −X −Y | {z } | {z } A(X,Y,Z) A(X,Y,I)       −YI 0 I 0 0 0 I 0  X 0 I  ·  YI 0  =  0 0 I  0 −X −Y −X 0 I [Y, X] −X −Y | {z } A(X,Y,I)

7 1.2.2 The homogenized Weyl algebra In [11] Berest and Wilson gave a new proof of (1.1) this using noncommutative algebraic geometry [6, 30]. That an approach based on noncommutative geometry should be possible was in fact anticipated very early by Le Bruyn who in [17] already came very close to proving (1.1). Let us indicate which methods are used.

Putting the standard Bernstein ﬁltration on the ﬁrst Weyl algebra A1

α β (A1)l = Spank{x y | α + β ≤ l} we ﬁnd that the Rees algebra corresponding to this ﬁltration is the so-called “homogenized Weyl algebra”

H = khx, y, zi/(zx − xz, zy − yz, xy − yx − z2) which is a noetherian connected k-algebra of global dimension three. Further, H is Koszul i.e. the minimal resolution for kH is of the form

3 3 0 → H(−3) → H(−2) → H(−1) → H → kH → 0 which is of the same form as for the commutative polynomial algebra A = k[x, y, z].

The relation A1 = H/(z − 1)H gives a close interaction between right A1- modules and graded right H-modules:

mod A1 → grmod H : M 7→ Rees module of M −1 grmod H → mod A1 : N 7→ N[z ]0 and under this correspondence we have

R(A1) ↔ R(H) = { reﬂexive rank one graded right H-modules }/ iso, shift

2 Following Artin and Zhang [6] we may associate to H a projective scheme Pq, which is essentially the quotient category of the ﬁnitely generated graded right H-modules grmod H by the full subcategory of ﬁnite dimensional H-modules tors H tails H := grmod H/ tors H

Thus we may consider A1 as the coordinate ring of an open aﬃne part of a 2 noncommutative space Pq with homogeneous coordinate ring” H. Also, the equality H/zH = k[x, y] gives rise to a restriction functor i∗ : tails H → coh P1. We now have

2 ∗ 1 R(H) ↔ R(Pq) = { reﬂexive rank one objects I ∈ tails A s.t. i I = OP }/ iso 2 Thus this leads to describing line bundles on the noncommutative plane Pq which 2 are framed at the the (commutative) line at inﬁnity. An object I ∈ R(Pq) is

8 now determined by a representation M of the quiver ∆q, which is the same as the earlier quiver ∆ except the relations now reﬂect the relations in H:

0 −→X −→X 0 1 2 Y 1 2 Y 1 2 H (Pq, I(−2)) −→ H (Pq, I(−1)) −→ H (Pq, I) 0 −→Z −→Z where  −YZ 0  0 0 0 X Y Z ·  X 0 Z  = 0 −Z −X −Y | {z } H(X,Y,Z) In addition, dimM = (n, n, n − 1) and the maps Z is an isomorphism and Z0 is surjective. As before the representation M is determined by the three linear maps on the left. The result is

a 0 R(A1) ↔ {(X,Y,Z) ∈ rep(n,n) ∆ | Z iso and rank H(X,Y,Z) ≤ 2n + 1}/ Gln(k) n where  −YXZ  H(X,Y,Z) =  Z 0 −X  0 Z −Y

And indeed, putting X = Z−1X, Y = Z−1Y gives a 0 = {(X, Y, I) ∈ rep(n,n) ∆ | rank H(X, Y, I) ≤ 2n + 1}/ Gln(k) n a 0 = {(X, Y, I) ∈ rep(n,n) ∆ | rank([Y, X] − I) ≤ 1}/ Gln(k) n a = Cn n where one uses  −1      Z 0 0 −YZ 0 −YI 0  0 Z−1 0  ·  X 0 Z  =  X 0 I  0 0 Z−1 −Z −X −Y −I −X −Y | {z } | {z } H(X,Y,Z) H(X,Y,I)       −YI 0 I 0 0 0 I 0  X 0 I  ·  YI 0  =  0 0 I  −I −X −Y −X 0 I [Y, X] − I −X −Y | {z } H(X,Y,I)

9 1.2.3 Three dimensional Sklyanin algebras We now observe that there are many other noncommutative algebras which give rise to a noncommutative projective planes than just the one associated to the Weyl algebra (this is in fact a fairly degenerate one) [3, 4, 10]. A generic class of them are the so-called Sklyanin algebras. A three dimensional Sklyanin algebra is a graded k-algebra

Skl = Skl3(a, b, c) = khx, y, zi/(f1, f2, f3) where f1, f2, f3 are the quadratic equations  2 f1 = ayz + bzy + cx  2 f2 = azx + bxz + cy 2  f3 = axy + byx + cz where a, b, c ∈ k outside a ﬁnite (known) set. The algebras Skl = Skl3(a, b, c) are so-called elliptic quantum polynomial rings. They correspond to the Koszul Artin-Schelter algebras of global dimension three where, in the associated geometric data, E is a smooth elliptic curve and σ is given by translation under the group law.

Assume that σ has inﬁnite order (this corresponds to the generic case). We now consider R(Skl) = { reﬂexive rank one graded right Skl-modules }/ iso, shift There is a restriction functor i∗ : tails Skl → coh E, leading to

2 R(Skl) ↔ R(Pq) = { reﬂexive rank one objects I ∈ tails Skl s.t. i∗I = vector bundle on E of degree 0 }/ iso Again by generalized Beilinson equivalence, the outcome is a R(Skl) ↔ Dn n where

0 Dn = {M = (X,Y,Z) ∈ rep(n,n) ∆ | M ⊥ V and

rank Skl(X,Y,Z) ≤ 2n + 1}/ Gln(k) Here V is a ﬁxed representation of ∆0 corresponding to a line bundle on E with i dimension vector dimV = (6, 3). Further, M ⊥ V means Ext∆0 (M,V ) = 0 for i ≤ 1. And  cX aZ bY  Skl(X,Y,Z) =  bZ cY aX  aY bX cZ We were able to show (see Theorem 5.7)

10 Theorem 1.3. The varieties Dn are smooth, connected and aﬃne varieties of dimension 2n.

Remark 1.4. 1. We would like to think of the varieties Dn as elliptic Calogero- Moser spaces. In particular D0 is a point and D1 is the complement of the elliptic curve E under a natural embedding in P2 (see Theorem 5.7 combined with Corollary 5.2). 2. In [22] Nevins and Staﬀord prove a more general result, for all three dimen- 2 sional Koszul Artin-Schelter regular algebras A. They construct Hilbn(Pq) as the scheme parameterising the torsion free graded right A-modules I of projective dimension one, up to shift of grading. Thus

2 Hilbn(Pq) ↔ {torsion free graded right A-modules I, pd I = 1}/ iso, shift (6) 2 They proved that Hilbn(Pq) is a connected projective variety of dimension 2n.

In case of a three dimensional Sklyanin algebra A = Skl3(a, b, c) where σ 2 has infinite order, Dn is an open affine dense part of Hilbn(Pq) dimension 2n (as it corresponds to reflexive modules).

On the other hand, if A = k[x, y, z] is commutative then the usual Hilbert 2 scheme of points Hilbn(P ) parameterizes the objects in (6). In contrast, the set

R(A) = { reﬂexive rank one graded right A-modules }/ iso, shift

is empty for n > 0 as every reﬂexive ideal is up to shift and isomorphism, equal to A.

11 2 Preliminaries and basic tools

In this section we gather some basic tools and results used along the way. These are collected from [2, 3, 4, 6, 20, 24, 25, 26, 27, 28, 29].

We begin with the folloxing convention: Convention 2.1. Whenever XyUvw(··· ) denotes an abelian category then xyuvw(··· ) denotes the full subcategory of XyUvw(··· ) consisting of noetherian objects.

2.1 Connected graded algebras and Hilbert series

Let A = ⊕i∈ZAi be a Z-graded k-algebra. We say that A is connected if in addition Ai = 0 for all i < 0 and A0 = k. We write GrMod A for the category of graded right A-modules with mor- phisms the A-module homomorphisms of degree zero. Since GrMod A is an abelian category with enough injective objects we may deﬁne the functors n ExtA(M, −) on GrMod A as the right derived functors of HomA(M, −). It is convenient to write (for n ≥ 0)

n M n ExtA(M,N) := ExtA(M,N(d)); d∈Z Let M be a graded right A-module. For any integer n, deﬁne M(n) as the graded A-module that is equal to M with its original A action, but which is graded by M(n)i = Mn+i. We refer to the functor M 7→ M(n) as the n-th shift functor.

Let A be a noetherian connected graded k-algebra. The Hilbert series of M ∈ grmod A is the Laurent power series +∞ X i hM (t) = (dimk Mi)t ∈ Z((t)). i=−∞ This deﬁnition makes sense since A is right noetherian. As an immediate con- −l sequence, hk(t) = 1 and hM(l)(t) = t hM (t). Assume further that A has ﬁnite global dimension. Given a projective resolution 0 → P r → ... → P 1 → P 0 → M → 0 we have r X i hM (t) = (−1) hP i (t). i=0 Since A is connected, left bounded graded right A-modules are projective if and only if they are free hence isomorphic to a sum of shifts of A. So if we write

ri i M P = A(−lij) j=0

12 we obtain

r X i hM (t) = (−1) h ri (t) ⊕j=O A(−lij ) i=0 r ri X i X lij = (−1) t hA(t) i=0 j=0 | {z } qM (t) where qM (t) is the so-called characteristic polynomial of M. Thus we have the formula

−1 qM (t) = hM (t)hA(t) (7)

−l −1 Note that qM(l) = t qM (t), qA(t) = 1 and qk(t) = hA(t) .

2.2 Three dimensional Artin-Schelter regular algebras Now we come to the definition of regular algebras. Introduced by Artin and Schelter [2] in 1986, they may be considered as non-commutative analogons of polynomial rings. Definition 2.2. [2] A connected graded k-algebra A is called an Artin-Schelter regular algebra of dimension d if it has the following properties: (i) A has finite global dimension d; (ii) A has polynomial growth, that is, there exists positive real numbers c, e e such that dimk An ≤ cn for all positive integers n; (iii) A is Gorenstein, meaning there is an integer l such that

k(l) if i = d, Exti (k ,A) =∼ A A A 0 otherwise.

where l is called the Gorenstein parameter 1 of A. If A is commutative, then the condition (i) already implies that A is isomorphic to a polynomial ring with some positive grading. The following questions for an Artin-Schelter regular algebra A of dimension d are still open in general: 1. Is e + 1 = d, where e is the minimal choice in Deﬁnition 2.2(ii)? Or equivalently, is gkdim A = gl dim A? Here gkdim A stands for the Gelfand- Kirilov dimension of A, see §2.3.1. 2. Is A a domain? 3. Is A noetherian?

1 It is easy to see that l = deg qk(t)

13 The main goal is of course the classiﬁcation all Artin-Schelter regular algebras of dimension d. At this moment this is still unknow for d ≥ 4, but completely solved for d = 3: • If d = 1 then A = k[x].

• If d = 2 then there are two possibilities in case A is generated in degree one: Either A is a so-called quantum plane

khx, yi/(yx − λxy) where λ ∈ k \{0}

or A is the Jordan quantum plane

khx, yi/(yx − xy − x2)

In case A is not generated in degree one some other (known) types occur. Note that the algebra khx, yi/(yx) is not an Artin-Schelter regular algebra. Although it has global dimension two and polynomial growth (even gkdim A = 2), it does not satisfy the Gorenstein condition since 1 ExtA(kA,A) 6= 0. This algebra is also the only graded algebra of global dimension two and gk-dimension two which is not noetherian (see [2]). • If d = 3 then there also exists a complete classiﬁcation for Artin-Schelter regular algebras of dimension three [2, 3, 4, 27, 28]:

Theorem 2.3. The three dimensional Artin-Schelter regular algebras A can be classiﬁed. They are all left and right noetherian domains with Hilbert series of a weighted polynomial ring k[x, y, z].

In what follows we will restrict ourselves to three dimensional Artin-Schelter regular algebras A which are generated in degree one. As proved in [2], there are two possibilities: • The minimal resolution of k has the form

3 3 0 → A(−3) → A(−2) → A(−1) → A → kA → 0

which means that A is Koszul. Hence A has three generators and three deﬁning homogeneous relations in degree two. It follows that the Hilbert 3 series of A is given by hA(t) = (1 − t) , which is the same as that of the commutative polynomial algebra k[x, y, z] with standard grading. Such algebras A are called quantum polynomial rings in three variables. Since the relations have degree two we also refer to these algebras as quadratic three dimensional Artin-Schelter regular algebras. • The minimal resolution of k has the form

2 2 0 → A(−4) → A(−3) → A(−1) → A → kA → 0

14 Thus A has two generators and two deﬁning homogeneous relations in degree three. We now deduce 1 h (t) = A (1 − t)2(1 − t2)

which is the same as that of the commutative polynomial algebra k[x, y, z] with grading deg x = deg y = 1, deg z = 2. We refer to these algebras as cubic three dimensional Artin-Schelter regular algebras. Example 2.4. The commutative polynomial ring in three variables k[x, y, z] with standard grading is a quadratic three dimensional Artin-Schelter regular algebra. In contrast, the weighted polynomial ring k[x, y, z] where deg x = deg y = 1, deg z = 2 is not a quadratic or cubic Artin-Schelter regular algebra of dimension three since it is not generated in degree one. Example 2.5. Standard examples (and in fact fairly degenerate ones) are pro- vided from homogenizations of the ﬁrst Weyl algebra

A1 = khx, yi/(xy − yx − 1)

We may homogenize the relation of A1 in two ways: • Introduce a third variable z which commutes with x and y, and for which yx − xy − z2 = 0. Thus deg z = 1, and we obtain the quadratic three dimensional Artin-Schelter regular algebra from the introduction

2 H = Hq = khx, y, zi/(zx − xz, zy − yz, xy − yx − z )

which we refer to as the homogenized Weyl algebra. It is easy to see that H is the Rees algebra with respect to the standard Bernstein ﬁltration on A1. • Introduce a third variable z which commutes with x and y and for which xy − yx − z = 0. Thus deg z = 2 and we obtain the enveloping algebra of the Heisenberg algebra, which is a cubic three dimensional Artin-Schelter regular algebra

Hc = khx, yi/(xz − zx, yz − zy, xy − yx − z) = khx, yi/(yx2 − 2xyx + x2y, xy2 − 2yxy + y2x) = khx, yi/([x, [x, y]], [y, [y, x]])

It is easy to see that Hc is also the Rees algebra associated to some filtra- tion on the first Weyl algebra A1. ∼ ∼ Since H/(z − 1) = A1 = Hc/(z − 1) it is clear that in both cases the right ideals of A1 are closely related to the reflexive rank one right modules over H and Hc.

15 Example 2.6. The generic quadratic three dimensional Artin-Schelter regular algebras are called three dimensional Sklyanin algebras. They are of the form

Skl3(a, b, c) = khx, y, zi/(f1, f2, f3) where f1, f2, f3 are the quadratic equations

 2 f1 = ayz + bzy + cx  2 f2 = azx + bxz + cy (8) 2  f3 = axy + byx + cz and (a, b, c) ∈ P2 \ F where F is the set

2 3 3 3 3 3 3 3 3 F = {(a, b, c) ∈ P | abc = 0 or a = b = c or (3abc) = (a + b + c ) }. Note [2] that A is not a skew polynomial ring, i.e. the relations cannot be written in the form X xixj = cklxkxl (k,l)<(i,j) (lexicographic ordering). Example 2.7. The generic cubic three dimensional Artin-Schelter regular algebras are of the form khx, yi/(f1, f2) where f1, f2 are the cubic equations

2 2 3 f1 = axy + byxy + ayx + cx 2 2 3 (9) f2 = ayx + bxyx + axy + cy and (a, b, c) ∈ P2 \ F where F is the set

2 2 2 2 F = {(a, b, c) ∈ P | a = b = c } ∪ {(0, 0, 1), (0, 1, 0)}. For the sequel we will mostly restict ourselves to quadratic three dimensional Artin-Schelter regular algebras (i.e. quantum polynomial rings in three variables), although it is interesting to see what happens in the cubic case.

2.3 Modules over quadratic Artin-Schelter regular algebras Let A denote a quadratic three dimensional Artin-Schelter regular algebra. Thus −3 the Hilbert series of A is hA(t) = (1 − t) . We will further examine the Hilbert series of modules over A. As an exercise, one may look what happens in the cubic case.

16 2.3.1 Hilbert series and gk-dimension

−1 Let M ∈ grmod A. We expand the characteristic polynomial qM (t) ∈ Z[t, t ] in powers of (1 − t)

2 3 qM (t) = r + a(1 − t) + b(1 − t) + f(t)(1 − t) (10) where r, a, b ∈ Z and f(t) ∈ Z[t, t−1] is a Laurent polynomial. Needless to say that q0 (1) q00 (1) r = q (1), a = − M , b = M M 1! 2!

Multiplying both sides of (10) with hA(t), equation (7) implies r a b h (t) = + + + f(t) M (1 − t)3 (1 − t)2 (1 − t)

It was shown in [4] that the Gelfand-Kirilov dimension gk M of M may be com- puted as the order of the pole of hM (t) at t = 1. It follows that gkdim A = 3 and if M ∈ grmod A then 0 ≤ gkdim M ≤ 3 is an integer. In particular, M is finite dimensional over k if and only if gk M = 0, i.e. r = a = b = 0. We will refer to the integer r as the rank of M. More general, if M 6= 0 then the first nonvanishing coefficient of the expansion of hM (t) in powers of 1−t is called the multiplicity e(M) of M. By definition of Hilbert series, e(M) > 0 for M 6= 0.

−l It is an easy exercise to compute hM(l)(t): From qM(l)(t) = t qM (t) we ﬁnd

q00 (1) 1 q (1) = r, −q0 (1) = lr + a, M(l) = l(l + 1)r + la + b M(l) M(l) 2 2 and therefore r lr + a l(l + 1)r/2 + la + b h (t) = + + + t−lf(t) (11) M(l) (1 − t)3 (1 − t)2 (1 − t)

In particular we have shown that the rank, multiplicity en Gelfand-Kirilov dimension of M are invariant under shifting. We will see some special types of modules in the next paragraph.

2.3.2 Linear modules A linear module of dimension d over A is a cyclic A-module generated M in degree zero with Hilbert series (1 − t)−d. Clearly 0 ≤ d ≤ 3, gk M = d, e(M) = 1 and a linear module of dimension zero (resp. three) is isomorphic to k (resp. A). Linear modules of dimension one and two are respectively called point and line modules. They were classiﬁed in [3, 4]. Line modules are of the form 2 A/uA = S with u ∈ A1. Hence line modules correspond naturally to lines in P .

17 We now show how point modules were classiﬁed in [3, 4]. We write the relations of A as     f1 x  f2  = M ·  y  f3 z

We introduce auxiliary (commuting) variables x(l), y(l), z(l) (for l ∈ Z) and for a monomial m = a0 ··· an where ai ∈ {x, y, z} we deﬁne the multilinearization (0) (n) of m as me as a0 ··· an . We extend this operation linearly to homogeneous polynomials in the variables x, y, z. 2 2 ˜ Let Γ ⊂ P × P denote the locus of common zeros of the fi. It turns out that Γ is the graph of an automorphism σ of E = pr1(Γ), the locus of zeros of the multihomogenized polynomial det(Mf). If det(Mf) is not identically zero then E is a divisor of degree 3 in P2. We then say that A is elliptic. Otherwise, E is all of P2 and we call A linear in this case. The connection between E and point modules is as follows: Let P be a point module over A. Since dimk Pi = 1 for i ≥ 0 we may choose a basis ei for each k- P vector space Pi. Thus P = kei. Multiplication by the generators x, y, z ∈ A1 of A induce linear maps Pi → Pi+1. Thus   eix = ei+1αi eiy = ei+1βi for some αi, βi, γi ∈ k  eiz = ei+1γi

2 Now since P is generated in degree one it is not hard to see that (αi, βi, γi) ∈ P . Further, e0fi = 0 hence ((α0, β0, γ0), (α1, β1, γ1)) ∈ Γ and hence (α0, β0, γ0) ∈ E. This construction is reversible and deﬁnes a bijection between the closed points of E and the point modules over A. We will often write Np for the point module corresponding to a point p ∈ E. Note that Np(1)≥0 = Npσ . Example 2.8. Consider the commutative polynomial ring A = k[x, y, z]. Then it is easy to see that E = P2 and σ = id. Thus k[x, y, z] is a linear Artin-Schelter regular algebra. Example 2.9. Consider the homogenized Weyl algebra

A = H = khx, y, zi/(xy − yx − z2, zx − xz, zy − yz)

Then   −y0 x0 −z0 Mf =  z0 0 −x0  (12) 0 z0 −y0

3 2 hence det(Mf) = −z0 , thus E is the “triple” line z = 0 in P : the points (x, y, ) such that 3 = 0. Since det(Mf) is not identically zero, H is an elliptic Artin- Schelter regular algebra.

18 Using the aﬃne coordinates u = y/x, v = z/x in P2 it is easy to check that the automorphism σ is given by σ(1, u, ) = (1, u + 2, ). Note that in particular σ has inﬁnite order.

Example 2.10. Consider a three dimensional Sklyanin algebra A = Skl3(a, b, c). Then the equation of E is deﬁned by the equation (a3 + b3 + c3)xyz = abc(x3 + y3 + z3) It follows that E is a smooth elliptic curve and σ is given by translation by some point ξ ∈ E under the group law. Choosing the rational point (1, −1, 0) on E as the orgin then ξ = (a, b, c).

2.4 Quantum projective planes A standard construction of algebraic geometry associates to any connected graded k-algebra A which is commutative, a projective scheme Proj A. A well- known theorem by Serre says that the abelian category of coherent sheaves on Proj A is equivalent to the quotient of the abelian category of ﬁnitely generated graded A-modules by the Serre subcategory of ﬁnite dimensional modules. It was the insight of Artin and Zhang [6] that this quotient category makes sense for any noncommutative connected graded k-algebra A.

2.4.1 Projective schemes Let A be a noetherian connected graded k-algebra. We define the noncommutative projective scheme X = Proj A of A as the triple (Tails A, OX , s) where Tails A is the quotient category of GrMod A modulo the direct limits of finite dimensional objects, OX is the image of A in Tails A and s is the automorphism M 7→ M(1) (induced by the corresponding functor on GrMod A). Convention 2.1 fixes the meaning of grmod A, tors(A) and tails A. It is easy to see that tors(A) consists of the finite dimensional graded right A-modules, and tails A = grmod A/ tors(A). We write Qcoh(X) = Tails A and we let coh(X) be the noetherian objects in Qcoh(X). Below it will be convenient to denote objects in Qcoh(X) by script letters, like M. We write π : GrMod A → Tails A for the quotient functor. The right adjoint ω of π is given by ωM = ⊕nΓ(X, M(n)) where as usual Γ(X, −) = HomQcoh(X)(OX , −). If M,N ∈ grmod A then ∼ ∼ πM = πN in coh(X) ⇔ M≥n = N≥n in grmod A for some n explaining the word “tails”.

For simplicity we also write HomX instead of HomQcoh(X). If M ∈ Qcoh(X) then HomX (M, −) is left exact, so we may deﬁne its right derived functors n ExtX (M, −). We deﬁne the cohomology groups of M by n n H (X, M) := Ext (OX , M).

19 In case A is a quadratic three dimensional Artin-Schelter regular algebra we usu- 2 ally write = X = Proj A and O = O 2 . The Gorenstein property determines Pq Pq the full cohomology groups of O   Al if i = 0 i 2 ∼ H (Pq, O(l)) = 0 if i 6= 0, 2 (13)  Homk(A−l−3, k) if i = 2

2 and Pq has cohomology dimension two (see [6]):

n max{n ∈ N | H (X, −) 6= 0} = 2

2 Finally, for 0 6= M ∈ coh(Pq) we let dim M = gk M − 1 and e(M) = e(M) where M ∈ grmod A, πM = M. The image under π of a point module P (resp. 2 line module S) over A will be called a point object on Pq (resp. line object). In particular, dim O = 2, dim S = 1, dim P = 0 where S = πS and P = πP .

2.4.2 The Grothendieck group and the Euler form We recall the deﬁnition of these notions in a more general setting.

Let C be an Ext-finite k-linear abelian category. Assume that C has finite i global dimension, which means that there exists an n such that ExtC(A, B) = 0 for all A, B ∈ C and all i > n. The Grothendieck group K0(C) of C is the abelian group generated by all objects of C (we write [A] ∈ K0(C) for A ∈ C) and for which we define [A] − [B] + [C] = 0 for A, B, C ∈ C whenever there is a short exact sequence 0 → A → B → C → 0 in C. It is easy to see that the following map defines a bilinear form

χ : K0(C) × K0(C) → Z X i i ([A], [B]) 7→ χ(A, B) = (−1) dimk ExtC(A, B) i which we call the Euler form for C. Note that we write χ(A, B) instead of χ([A], [B]).

Now let us take a more speciﬁc situation where C = coh(X) for a noetherian connected graded k-algebra A of ﬁnite global dimension and X = Proj A. We write K0(X) for K0(coh(X)). The shift functor on coh(X) induces an automorphism of K0(X):

sh : K0(X) → K0(X) [M] 7→ [M(1)]

−1 We may view K0(X) as a Z[t, t ]-module with t acting as the shift functor −1 sh . In [20] it was shown that K0(X) may be described in terms of the Hilbert

20 series of A: We have an isomorphism of Z[t, t−1]-modules

∼= −1 θ : K0(X) −→ Z[t, t ] /(qk(t)) (14) [M] 7→ qM (t) where M ∈ grmod A, πM = M.

In particular, [O(n)] is sent to t−n.

2.4.3 Serre duality Let A be a quadratic three dimensional Artin-Schelter regular algebra. We have an analogous form of the Serre duality on P2 (see [13]): b 2 Theorem 2.11 (Serre Duality). Let M, N ∈ D (coh(Pq)). Then there are natural isomorphisms

i ∼ 2−i ∗ ExtDb(coh( 2 ))(M, N ) = Ext b 2 (N , M(−3)) Pq D (coh(Pq))

b 2 Corollary 2.12. Let M ∈ D (coh(Pq)). For p ∈ E, denote the corresponding 2 point module by Np ∈ grmod A and Np = πNp ∈ coh(Pq) for the associated point object. Then

i ∼ 2−i ∗ ExtDb(coh( 2 ))(M, Np) = Ext b 2 (N σ3 , M) Pq D (coh(Pq)) p Proof. By Theorem 2.11 we have

i ∼ 2−i ∗ ExtDb(coh( 2 ))(M, Np) = Ext b 2 (Np(3), M) Pq D (coh(Pq)) and it follows from Np(1)≥0 = Npσ that ∼ Np(3) = πNp(3) = πNpσ3 = Npσ3 ends the proof.

21 3 From reﬂexive modules to the quantum plane

In this section A will be a quadratic three dimensional Artin-Schelter regular 2 2 algebra. We write Pq = Proj(A), πA = O and coh(Pq) = tails A.

An A-module M ∈ grmod A is said to be reﬂexive if M ∗∗ = M where ∗ M = HomA(M,A) is the dual of M. Our aim is to relate reﬂexive A-modules 2 M with certain objects M on Pq (which we call vector bundles). In the rank one case we will see that there is a natural shift l such that M(l) has a nice 2 presentation in the Grothendieck group K0(Pq). We then refer to M(l) as a normalized line bundle. In particular we compute partially the cohomology of 2 normalized line bundles on Pq.

3.1 Reﬂexive modules and vector bundles 2 A nonzero object in grmod A or coh(Pq) is called torsionif it has rank zero and it is called torsion free if it contains no torsion subobject. The following lemma helps us to characterise these objects. Proofs are found by using [4, Proposition 2.40 and Corollary 4.2]. Lemma 3.1. For 0 6= M ∈ grmod A the following are equivalent 1. M is torsion free 2. the canonical morphism µ : M → M ∗∗ is injective

3. HomA(N,M) = 0 for all N ∈ grmod A of gk-dimension ≤ 2. 2 and for 0 6= M ∈ coh(Pq) the following are equivalent 1. M is torsion free

2 2. Hom 2 (N , M) = 0 for all N ∈ coh( ) of dimension ≤ 1. Pq Pq 2 2 An object M ∈ coh(Pq) is called reflexive (or a vector bundle on Pq) if 2 M = πM for some reflexive object M ∈ grmod A. Vector bundles on Pq of rank 2 one are called line bundles on Pq. Lemma 3.2. For 0 6= M ∈ grmod A the following are equivalent 1. M is reflexive i.e. the canonical morphism µ : M → M ∗∗ is an isomorphism

1 2. M is torsion free and ExtA(N,M) = 0 for all N ∈ grmod A of gk- dimension ≤ 1.

2 and for 0 6= M ∈ coh(Pq) the following are equivalent 2 1. M is a vector bundle on Pq

1 2 2. M is torsion free and Ext 2 (N , M) = 0 for all N ∈ coh( ) of dimen- Pq Pq sion 0.

22 The following proposition shows the relationship between the objects we 2 defined in grmod A and coh(Pq). Proposition 3.3. The functors π and ω define inverse equivalences between 2 the full subcategories of grmod A and coh(Pq) with objects { torsion free objects in grmod A of projective dimension one } and 2 { torsion free objects in coh(Pq)} which restricts to inverse equivalences π and ω between the full subcategories of 2 grmod A and coh(Pq) with objects { reflexive objects in grmod A} and 2 { vector bundles on Pq} Proof. See [14, Corollary 3.4.2].

2 3.2 The Grothendieck group and the Euler form for Pq 2 We will introduce a natural Z-module basis for the Grothendieck group K0(Pq) and determine the matrix representation of the Euler form χ with respect to this natural basis.

2 Let M ∈ coh(Pq). Thus M = πM for some M ∈ grmod A. Recall from −1 (10) the expansion of the characteristic polynomial qM (t) ∈ Z[t, t ] in powers of (1 − t) 2 3 qM (t) = r + a(1 − t) + b(1 − t) + f(t)(1 − t) where r, a, b ∈ Z and f(t) ∈ Z[t, t−1] is a Laurent polynomial. Now let P be a point module and S a line module over A. Denote the corresponding objects in 2 coh(Pq) by P and S. Since

2 q(A)(t) = 1, qS(t) = 1 − t, qP (t) = (1 − t) it follows from (14) that

[M] = r[O] + a[S] + b[P]

2 hence {[O], [S], [P]} is a Z-module basis for K0(Pq) (which does not depend on the particular choice of P and S). Also, e(M) is the ﬁrst (leftmost) nonvanishing coordinate of [M] with respect to this basis. It follows from (11) that 1 [M(l)] = r[O] + (lr + a)[S] + l(l + 1)r + la + b [P] (15) 2 for all integers l.

23 2 Let us ﬁx such a basis {[O], [S], [P]} for K0(Pq). From the cohomology groups of O (see (13)) we deduce χ(O, O(l)) = (l + 1)(l + 2)/2 and using the resolutions for S, P 0 → O(−1) → O → S → 0, 0 → O(−2) → O(−1)2 → O → P → 0

2 one easily veriﬁes that the matrix representation of the Euler form χ for coh(Pq) is given by  1 1 1   −2 −1 0  (16) 1 0 0

3.3 Normalized rank one modules and sheaves Let I ∈ grmod A have rank one. It follows from (11) that there is an unique shift l such that the Hilbert series of I(l) has the form 1 n h (t) = − + f(t) I(l) (1 − t)3 1 − t for some n ∈ Z and f(t) ∈ Z[t, t−1]. This is equivalent with

dimk Am − dimk I(l)m = n for m 0 and due to §3.2 we also have [πI] = [O] − n[P]. We say that I(l) is normalized, 2 and has invariant n. Similary, an object I ∈ coh(Pq) of rank one there is an unique shift l such that [I(l)] = [O] − n[P] for some integer n. In that case we also refer to I(l) as normalized and call n the invariant. We will prove later (Theorem 3.7) that this invariant n is actually positive.

It is easy to see that Proposition 3.3 restricts to Proposition 3.4. The functors π and ω deﬁne inverse equivalences between 2 the full subcategories of grmod A and coh(Pq) with objects

{ normalized torsion free rank one objects in grmod A of projective dimension one and invariant n } (17) and

2 { normalized torsion free rank one objects in coh(Pq) with invariant n} which restricts to inverse equivalences π and ω between the full subcategories of 2 grmod A and coh(Pq) with objects

Rn(A) = { normalized reﬂexive rank one objects in grmod A with invariant n} and 2 2 Rn(Pq) = { normalized line bundles on Pq with invariant n}

24 Remark 3.5. 1. It turns out (see [22]) that the correct generalisation of the 2 2 Hilbert scheme of points on P is to deﬁne Hilbn(Pq) as the scheme parameterising the objects in (17). It is easy to see that if A is commutative then this condition singles out precisely the graded A-modules which occur as the graded ideal IX for a zero dimensional subscheme X of degree 2 2 n in P , i.e. X ∈ Hilbn(P ).

2. Observe that a nonzero morphism in the category Rn(A) is an isomor- 2 phism. Thus Rn(A) and Rn(Pq) are in fact groupoids. 3. We obtain a natural one to one correspondences between the elements of the set R(A) = { reﬂexive rank one graded right A-modules }/ iso, shift

` ` 2 and the isomorphism classes in the categories n Rn(A) and n Rn(Pq). 0 Example 3.6. Consider for two elements 0 6= l, l ∈ A1 the following map of right A-modules given by matrix multiplication

 l  ·  l0  A(−2) −−−−−−→ A(−1)2 Since A is a domain, this map is injective. Writing I for the cokernel we have an exact sequence of the form 0 → A(−2) → A(−1)2 → I → 0 which is in fact a minimal resolution of I since the matrix entries have positive degree. Thus I has rank one. Calculation of the Hilbert series of I yields 2t − t2 1 1 h (t) = h 2 (t) − h (t) = = − I A(−1) A(−2) (1 − t)3 (1 − t)3 1 − t = 0 + 2t + 5t2 + 9t3 + 14t4 + 20t5 + ... We either have • l(p) = l0(p) = 0 for some p ∈ E. If furthermore l, l0 are lineary independent over k then there exists a point module Nq over A such that (see [1, (2.6)])

0 → I → A → Nq → 0 Thus I is torsion free. We conclude that I has rank one, is torsion free, projective dimension one and normalized and has invariant one. Note I is not reﬂexive since I ⊂ A thus I∗∗ = A 6= I. If A is a three-dimensional 2 Sklyanin algebra we have q = pσ and if A = k[x, y, z] then q = p. • There is no p ∈ E such that l(p) = l0(p) = 0 (and as a consequence l, l0 are lineary independent over k) then one easily deduces that I is reﬂexive. Thus I ∈ R1(A). Note that this situation cannot occur in case A = k[x, y, z].

25 3.4 Cohomology of line bundles on the quantum plane We will restrict our attention to the reﬂexive rank one modules. It follows from Proposition 3.4 that it is equivalent to describe all normalized line bundles on 2 Pq. In the next theorem we compute partially the cohomology of such line bundles. Athough the result may be extended to normalized torsion free rank 2 one objects on Pq. ∼ 2 Theorem 3.7. Let I= 6 O be a normalized line bundle on Pq with invariant n. Thus I ∈ Rn. Then

0 2 1. H (Pq, I(l)) = 0 for l ≤ 0, 2 2 H (Pq, I(l)) = 0 for l ≥ −3; 1 2 2. dimk H (Pq, I) = n − 1 1 2 dimk H (Pq, I(−1)) = n 1 2 dimk H (Pq, I(−2)) = n 1 2 dimk H (Pq, I(−3)) = n − 1. As a consequence, n is positive and nonzero.

Proof. First let l ≤ 0. Suppose f is a nonzero morphism in Hom 2 (O, I(l)). Pq Since O and I(l) are both torsion free it is easy to see that f is injective and from

0 → O → I(l) → coker f → 0 (18) we get [coker f] = l[S] + (l(l + 1)/2 − n)[P]. By I 6=∼ O, coker f 6= 0. Hence l ≥ 0 (otherwise e(coker f) = l < 0 which is impossible) thus l = 0 and [coker f] = −n[P]. We obtain dim(coker f) = 0. By Lemma 3.2 the exact sequence (18) splits hence I is not torsion free. A contradiction. We conclude that Hom 2 (O, I(l)) = 0 for l ≤ 0. Pq Further, Serre duality (Theorem 2.11) yields

2 ∗ ∼ Ext 2 (O, I(l)) = Hom 2 (I(l + 3), O) Pq Pq

2 and by an same reasoning as above we ﬁnd Ext 2 (O, I(l)) = 0 for l ≥ −3. Pq For the second part, using (15) and (16) we obtain 1 χ(O, I(l)) = (l + 1)(l + 2) − n (19) 2 for all integers l. In particular, if −3 ≤ l ≤ 0 the ﬁrst statement gives

0 2 1 2 2 2 χ(O, I(l)) = dimk H (Pq, I(l)) − dimk H (Pq, I(l)) + dimk H (Pq, I(l)) 1 2 = − dimk H (Pq, I(l)) and comparing with the expression (19) completes the proof.

26 4 From the quantum plane to the elliptic curve

Let A be a quadratic three dimensional Artin-Schelter regular algebra.

4.1 Geometric data associated to quantum planes As seen above, there is a triple (E, σ, L) associated to A where

• Either j : E =∼ P2 or j : E,→ P2 is a divisor of degree 3 in P2, • σ ∈ Aut(E).

∗ 2 •L = j OP (1) an invertible OE-module Having this data we can try to recover A as a quotient of the tensor algebra 0 on H (E, L) = A1 by the ideal generated by the tensors f ∈ A1 ⊗ A1 whose multilinearisations f˜ vanish on Γ = (E, σ(E)). As shown in [3], this leads to the so-called “twisted” homogeneous coordinate ring B. We recall this procedure. More details can be found in [3, 5].

⊗n Write A = T/I where T = A1 = khx1, x2, x3i. Let f ∈ Ad and write ˜ P the multilinearisation of f as f = a0a1 . . . ad. Under the identiﬁcation A1 = H0(E, L) this induces a map of k-vector spaces

0 0 0 σ 0 σd−1 ρ : Td → H (E, L) ⊗k H (E, L ) ⊗k · · · ⊗k H (E, L ) X σ σd−1 f 7→ a0 ⊗k a1 ⊗k · · · ⊗k ad−1 where aσ := a ◦ σ is now a global section of the pullback Lσ := σ∗L of L under σ. We compose ρ0 with the natural map

0 0 σ 0 σd−1 0 H (E, L) ⊗k H (E, L ) ⊗k · · · ⊗k H (E, L ) → H (E, Ld) =: Bd

σ σd−1 where Ld := L ⊗E L ⊗E · · · ⊗E L to obtain a map of graded k-vector spaces ρ : T → B where B = ⊕dBd. Note that

0 0 0 2 2 B0 = H (E, OE) = k, B1 = H (E, L) = H (P , OP (1)) = A1

The advantage is that ρ(f) = 0 whenever f ∈ Id is a relation of A. It is easy to see that for ρ to extend to a k-algebra map, we have to deﬁne multiplication on

B as m m σ σ m a · b = µm,n(a ⊗ b ) where b = b ◦ σ for a ∈ Bm, b ∈ Bn, where µm,n is the natural map

0 0 σm 0 µm,n : H (E, Lm) ⊗k H (E, Ln ) → H (E, Lm+n) Since I is in the kernel of the homomorphism ρ : T → B it factors through T/I = A. Clearly ρ is bijective in degree one. In [3] it is shown that ρ is in fact surjective and

27 • If A is linear then A =∼ B • If A is elliptic then A/gA =∼ B where g is a regular normalizing element of degree three. All point modules are B-modules. In other words, if A is elliptic then g annihi- lates all point modules: Np · g = 0 for all p ∈ E. We have seen that the stucture 0 sheaf OE on E corresponds to the graded k-algebra B = ⊕dH (E, OE ⊗E Ld). We may extend this operation to any sheaf M ∈ Qcoh E by putting

0 Γ∗(M) = ⊕nH (E, M ⊗E Ln) and we deﬁne Ln for n < 0 as

σn σ−2 σ−1 Ln = L ⊗E · · · ⊗E L ⊗E L

σ−1 where L = σ∗L. It is clear that Γ∗(M)≥0 is a right B-module by the natural map

0 0 0 σn Γ∗(M)m⊗kΓ∗(OE)n = H (E, M⊗ELn)⊗kH (E, Lm) → H (E, M⊗ELn⊗ELm )

It follows from [6, 5] that the functor Γ∗ : Qcoh E → GrMod B deﬁnes an equivalence Qcoh E =∼ Tails B. The inverse of this equivalence and its composition with π : GrMod B → Tails B are both denoted by (g−). Composition with the map ρ : A → B yields functors

∗ i πM = (M ⊗A B)e i∗M = π(Γ∗(M)A)

We will call i∗(πM) the restriction of πM to E. We may sketch these functors in a commutative diagram

− ⊗A B GrMod A GrMod B (g−) (−)A ωπ ωπ Γ∗

i∗ (g−) 2 Qcoh Pq Tails B Qcoh E

Γ∗ i∗

∗ Note that i∗ is exact. For the left derived functor of i we have

28 L − ∗ Lemma 4.1.1. 1. If M ∈ D (GrMod A) then Li (πM) = (M ⊗A B)e. 2. In particular, if A is elliptic and M ∈ grmod A then

∗ A (a) Lji (πM) = (Torj (M,B))e A (b) Torj (M,B) = 0 for j ≥ 2. A g· (c) Tor1 (M,B) = 0 if and only if M is g-torsionfree i.e. M(−3) −→ M is injective.

Idea of the proof for part (2): Applying M⊗A− to the exact sequence in grmod A

g· 0 → A(−3) −→ A → B → 0 we ﬁnd A g· 0 → Tor1 (M,B) → M(−3) −→ M → M ⊗A B → 0

Remark 4.1. Although we will mostly assume that A is a three dimensional Sklyanin algebra it is interesting to see how the obtained results have an analogue for other quadratic Artin-Schelter regular algebras such as the commutative polynomial ring k[x, y, z] or the homogenized Weyl algebra H. In this perspective, the fact that A may be linear or elliptic presents notational problems (see also [14]). Also E may be non-reduced (for example in case A is the homogenized Weyl algebra). We may side step these problems by deﬁning E0 as

0 • E = Ered if A is elliptic,

• E0 is a σ invariant line in P2 if A is linear. In case A = k[x, y, z] we prefer E0 to be the line z = 0. There is an in- clusion j0 : E0 ,→ P2 and the geometric data (E, σ, L) then restricts to ge- 0 0 0 0 0 ometric data (E , σ , L ) where σ = σ|E0 is the restriction of σ to E and 0 0∗ 2 L = j OP (1). Similar as above one constructs the twisted homogeneous coordinate ring B0 = B(E0, σ0, L0) associated to the triple (E0, σ0, L0). As shown in [4, Proposition 5.13] there is a surjective map A → B0 whose kernel is generated by a normalizing element g0. In case A = k[x, y, z] or A is the homogenized Weyl algebra we now have that σ0 = id and g0 = z, hence in both cases B0 = k[x, y] is just the commutative polynomial ring in two variables. While if A is a three dimensional Sklyanin algebra we can take g0 = g. Further, there is still an 0 0 0∗ 0 equivalence Tails B = Qcoh E and analogous functors i , i∗ are deﬁned to obtain a similar commutative diagrams and results as above.

29 4.2 Restriction of line bundles to the elliptic curve

In this part we assume that A is a three dimensional Sklyanin algebra Skl3(a, b, c). Thus E is a smooth elliptic curve.

∼ Recall from [15, Ex. II, 6.11] that K0(E) = Z⊕Pic(E), where the projection rank : K0(E) → Z is given by the rank and the projection c1 : K0(E) → Pic(E) is the ﬁrst Chern class. For p ∈ E we have c1(Op) = OE(p). There is a homomorphism deg : Pic(E) → Z which assigns to a line bundle its degree. This extends (see [15, Ex. II, 6.12]) to a homomorphism deg : K0(E) → Z by assign- ing to a ﬁnite length object F ∈ coh E its length.

∗ 2 The functor i : coh Pq → coh E induces a group homomorphism ∗ 2 i : K0(Pq) → K0(E) X j ∗ ∗ ∗ [M] 7→ (−1) [Lji M] = [i M] − [L1i M] j One computes ∗ i [O] = [OE] ∗ i [S] = [Ou] + [Ov] + [Ow] u, v, w arbitrary but colinear ∗ i [P] = [Op] − [Opσ−3 ] p ∈ E arbitrary Hence if [M] = a[O] + b[S] + c[P] then rank i∗[M] = a = rank M deg i∗[M] = 3b 2 ∗ Theorem 4.2. 1. If I is a line bundle on Pq then i I is a line bundle on E. In particular, I is normalized if and only if i∗I has degree zero. In ∗ that case, c1(i I) = O((o) − (3nξ)) where n is the invariant of I. b 2 2. Assume that σ has inﬁnite order and that M ∈ D (coh Pq) is such that ∗ 2 2 Li M is a line bundle on E. Then M ∈ coh Pq is a line bundle on Pq. In particular, 2 2 ∗ Rn(Pq) = {M ∈ coh Pq | i M ∈ coh E is a line bundle of degree zero } Proof. See [13, Propositions 4.3 and 4.4]. Remark 4.3. If A = H is the homogenized Weyl algebra, we obtain a similar 0 1 result (by replacing E with E = Ered = P ), stated in [11] 2 ∗ 1 • If I is a line bundle on Pq then i I is a line bundle on P , •I is normalized if and only if i∗I has degree zero, i.e. if and only if ∗ 1 ∼ 1 i I = OP (since Pic(P ) = Z). Thus 2 2 ∗ 1 Rn(Pq) = {M ∈ coh Pq | i M ∈ coh P is a line bundle of degree zero } 2 ∗ ∼ 1 = {M ∈ coh Pq | i M = OP }

30 5 From the quantum plane to quiver representations

Throughout this Section 5, A is a quadratic three dimensional Artin-Schelter regular algebra. Although we are mostly dealing with three dimensional Sklyanin algebras, there are some results which hold more generally. In view of any possible confusion, we will repreat as much as possible which algebra A we are dealing with.

5.1 Generalized Beilinson equivalence

Consider the (commutative) projective plane P2. We have an equivalence of derived categories, known as Beilinson equivalence (see [8])

RHom 2 (E,–) b 2 −→P b D (coh P ) ←− D (mod ∆) L – ⊗∆E

2 2 2 where E = OP (2)⊕OP (1)⊕OP and mod ∆ is the category of ﬁnite dimensional representations of the quiver ∆ (representations are always assumed to satisfy the relations)

X X −→−2 −→−1 Y Y −2 −→−2 −1 −→−1 0 Z Z −→−2 −→−1 with relations reﬂecting the relations in the polynomial algebra k[x, y, z]   Y−2Z−1 = Z−2Y−1 Z−2X−1 = X−2Z−1  X−2Y−1 = Y−2X−1

Now let A be a quadratic three dimensional Artin-Schelter regular algebra. Then we have a similar situation: There is an equivalence of derived categories (obtained in a similar way as in [9, Theorem 6.2])

RHom 2 (E,–) Pq b 2 −→ b D (coh Pq) ←− D (mod ∆) (20) L – ⊗∆E where E = O(2) ⊕ O(1) ⊕ O and ∆ is the quiver

X X −→−2 −→−1 Y Y −2 −→−2 −1 −→−1 0 Z Z −→−2 −→−1

31 with relations reﬂecting the relations of A. For example, if A = H is the homogenized Weyl algebra then   Z−2X−1 = X−2Z−1 Z−2Y−1 = Y−2Z−1  X−2Y−1 − Y−2X−1 = Z−2Z−1

In case A = Skl3(a, b, c) is a three dimensional Sklyanin algebra then   aY−2Z−1 + bZ−2Y−1 + cX−2X−1 = 0 aZ−2X−1 + bX−2Z−1 + cY−2Y−1 = 0  aX−2Y−1 + bY−2X−1 + cZ−2Z−1 = 0

Let us just recall how the equivalence (20) works. Let

2 M D = Hom 2 (E, E) = Hom 2 (O(i), O(j)) Pq Pq i,j=0 the algebra of endomorphisms of E. We consider the left exact functor Hom 2 (E, −) Pq 2 which takes coherent sheaves on to right D-modules. Now Hom 2 (E, −) ex- Pq Pq tends to a functor RHom 2 (E, −) on bounded derived categories. On the other Pq hand, it is easy to see that D =∼ k∆/(R). Since the category Mod ∆ of representations of ∆ is equivalent to the category of right k∆/(R)-modules we deduce Mod ∆ =∼ Mod D.

For a non-negative integer i the equivalence (20) restricts to an equivalence 2 between Xi and Yi where Xi ⊂ coh Pq is the full subcategory with objects

2 j Xi = {M ∈ coh q | Ext 2 (E, M) = 0 for j 6= i} P Pq and Yi ⊂ mod ∆ the full subcategory with objects

∆ Yi = {M ∈ mod ∆ | Torj (M, E) = 0 for j 6= i}.

i The inverse equivalences between these categories are given by Ext 2 (E, −) and Pq ∆ Tori (−, E):

1 Ext 2 (E,–) Pq −→ Xi ←− Yi (21) ∆ Tori (−,E)

32 5.2 The Grothendieck group and the Euler form for ∆ For a representation M ∈ mod ∆ we will denote the dimension vector of M as

3 dimM = (dimk M−2, dimk M−1, dimk M0) ∈ Z Since dim is exact on short exact sequences, it extends to a group morphism

3 ϕ : K0(∆) → Z where K0(∆) stands for the Grothendieck group K0(mod ∆) of mod ∆. We write Si for the simple representation of ∆ corresponding to the vertex i. Thus dimS0 = (0, 0, 1) etc. Since the image of [S−2], [S−1], [S0] under ϕ is a basis on the right, we conclude that ϕ is an isomorphism and {[S−2], [S−1], [S0]} is a basis for K0(∆). We will ﬁx this basis, and in what folows we will often identify 3 K0(∆) = Z .

It is also standard that the matrix representation of the Euler form χ : K0(∆) × K0(∆) → Z with respect to this basis is given by 1 −3 3  0 1 −3 (22) 0 0 1

∼ 3 Under the above isomorphism K0(∆) = Z we identify χ with the associated bilinear form on Z3.

2 5.3 First description of Rn(Pq) Let A be a quadratic three dimensional Artin-Schelter regular algebra.

2 We would like to understand the image of Rn(Pq) under the generalized 2 2 Beilinson equivalence (20). Recall that Rn(Pq) is the full subcategory of coh Pq which objects are given by

2 2 Rn(Pq) = { normalized line bundles on Pq with invariant n } 2 = {M ∈ coh Pq | M reﬂexive and [M] = [O] − n[P]}

2 b 2 Let M be an object of Rn(Pq) and consider M as a complex in D (coh Pq) of degree zero. Then Theorem 3.7 implies that M ∈ X1, thus the image of this complex is concentrated in degree one

RHom 2 (E, M) = M[−1] Pq

1 where M = Ext 2 (E, M). Hence M is a representation of ∆. By functoriality, Pq multiplication by x ∈ A induces linear maps

1 2 M(X−2) 1 2 M(X−1) 1 2 H (Pq, M(−2)) −→ H (Pq, M(−1)) −→ H (Pq, M)

33 and similar for multiplication with y, z hence M is determined by the following representation of ∆

M(X ) M(X ) −→−2 −→−1 1 2 M(Y−2) 1 2 M(Y−1) 1 2 H (Pq, M(−2)) −→ H (Pq, M(−1)) −→ H (Pq, M) M(Z ) M(Z ) −→−2 −→−1

Theorem 3.7 implies that dimM = (n, n, n − 1). We now investigate how the reﬂexivity of M is translated through the derived equivalence (20). By Lemma 3.2, we have

i 2 Ext 2 (N , M) = 0 for i ≤ 1 and all N ∈ coh , dim N = 0 (23) Pq Pq

2 2 Point objects on Pq are simple zero-dimensional objects in coh Pq. Note that if the automorphism σ has inﬁnite order, the converse is also true (see [4]). Hence (23) implies (and if order(σ) = ∞, is equivalent with)

i 2 Ext 2 (P, M) = 0 for i ≤ 1 and all point objects P on Pq Pq

2 Using the Euler form on Pq, we obtain χ(M, P) = 1 for all point objects P on 2 Pq. Thus

k[−2] = RHom 2 (P, M) =∼ RHom (RHom 2 (E, P),M[−1]) Pq ∆ Pq

Now O is reﬂexive and therefore P ∈ X0. Thus if we consider P as a complex b 2 in D (coh Pq) of degree zero then the image of this complex is concentrated in degree zero RHom 2 (E, P) = P Pq

σi where P = Hom 2 (E, P). If we denote P = N = πN where p = (α , β , γ ) ∈ Pq p p i i i E then the representation of ∆ corresponding to P (hence P) is given by

α α −→−2 −→−1 β β k −→−2 k −→−1 k γ γ −→−2 −→−1

We will denote this diagram also by p, thus we write P = p. We have found

RHom∆(p, M) = k[−1] for all p ∈ E

2 which implies Hom∆(p, M) = 0 and ExtD(p, M) = 0 for all p ∈ E. By Corollary 2.12 one obtains

2 2 2 Ext (p, M) = H (RHom (p, M)) = H (RHom 2 (P, M[1])) D D Pq 0 0 ∗ σ3 ∗ =∼ H (RHom 2 (M[1], P )) = Hom (M, p ) Pq ∆

34 hence Hom∆(M, p) = 0 for all p ∈ E. Thus if we let Cn(∆) be the image of 2 ∼ Rn(Pq) under the equivalence X1 = Y1 then 2 M ∈ Rn(Pq) ⇒ M ∈ Cn(∆) where

1 M = Ext 2 (E, M) ∈ mod ∆, dimM = (n, n, n − 1) Pq

and Hom∆(M, p) = 0, Hom∆(p, M) = 0 for all p ∈ E As we have more or less indicated, there is hope that these properties charac- 2 terise normalized line bundles on Pq in case σ has infinite order. The following theorem shows that this is true in the Sklyanin case. We conjecture that The- orem 5.1 is true for all quadratic Artin-Schelter regular algebras where σ has infinite order. Theorem 5.1. Let A be a three dimensional Sklyanin algebra where σ has infinite order. Let n > 0. Then there is an equivalence of categories

1 Ext 2 (E,–) Pq 2 −→ Rn(Pq) ←− Cn(∆) ∆ Tor1 (–,E) where

Cn(∆) = {M ∈ mod ∆ | dimM = (n, n, n − 1) and

Hom∆(M, p) = 0, Hom∆(p, M) = 0 for all p ∈ E}. Proof. The part has already been shown above. Conversely by Corollary 2.12

L 2 2 H (RHom (M, p)) = H (RHom 2 (M ⊗ E, P)) ∆ Pq ∆ L 0 0 0 0 =∼ H (RHom 2 (P ,M ⊗ E)) = H (RHom (p ,M)) Pq ∆ ∆

0 σ3 2 where p = p . Thus Hom∆(M, p) = Ext∆(M, p) = 0 for all p ∈ E. Now 1 gl dim D = 2 so we may compute dimk Ext∆(M, p) using the Euler form on 1 mod ∆. We obtain χ(p, M) = −1 hence Ext∆(M, p) = k. In other words RHom∆(M[−1], p) = k. L b 2 Put M = M[−1]⊗∆E. By the category equivalence (20) between D (coh Pq) b and D (mod ∆) we obtain RHom 2 (M, P) = k, giving (by adjointness) Pq ∗ RHomE(Li M, Op) = k. Since E is a smooth elliptic curve it is easy to see that this implies that Li∗M is a line bundle on E. Hence by Theorem 4.2 the same is true for M. In particular, M ∈ Y1. We only have to check that M is normalized. The derived equivalence gives rise to inverse group isomorphisms

2 µ : K0(Pq) → K0(∆) X i i [N ] 7→ (−1) [Ext 2 (E, N )] Pq i

35 and

2 ν : K0(∆) → K0(Pq) X i ∆ [N] 7→ (−1) [Tori (N, E)] i

2 The basis {[O], [S], [P]} for Pq now corresponds to the Z-basis {[S0], −2[S−2] − [S−1], [S−2]+[S−1]+[S0]} for K0(∆). And since M ∈ Y1 we have ν[M] = −[M]. It is easy to check that [M] = [O] − n[P]. We conclude that M is a normalized 2 line bundle on Pq.

5.4 Line bundles with invariant one Using the material from the previous section it is now easy to parametrize the 2 line bundles on Pq with invariant one in case A = Skl3(a, b, c).

Corollary 5.2. Let A = Skl3(a, b, c) be a three dimensional Sklyanin algebra for which σ has inﬁnite order. The representations in C1(∆) are the representations

−→α −→0 β k −→ k −→0 0 (24) γ −→ −→0 for some (α, β, γ) ∈ P2 − E

Proof. Follows from Theorem 5.1 and the description of Cn(∆).

5.5 Analogy with the homogenized Weyl algebra In order to point the analogy with the ﬁrst Weyl algebra, we recall some results from [11]. Some of them were already discussed in the introduction. Below, H is the homogenized Weyl algebra.

Let R(A1) be the category of right ideals of the ﬁrst Weyl algebra A1, with maps given by isomorphisms. As before, let R(H) be the category of normalized rank one reﬂexive graded right H-modules, with maps given by isomorphisms. Then there is an equivalence between these two groupoids ∼ R(A0) = R(H)

2 As in §3.3 if we write Rn(H) resp. Rn(Pq) for the full subcategory of grmod(H) 2 resp. coh Pq in which the objects are

Rn(H) = { normalized reﬂexive rank one graded right H-modules with invariant n } and 2 2 Rn(Pq) = { normalized line bundles on Pq }

36 then the functors π and ω deﬁne inverse equivalences between these categories (which are in fact groupoids) ∼ 2 Rn(H) = Rn(Pq) Thus ∼ ∼ a ∼ a 2 R(A0) = R(H) = Rn(H) = Rn(Pq) n n Using the generalized Beilinson equivalence (20) it was then shown in [11] that there is an equivalence of categories (for n > 0)

1 Ext 2 (E,–) Pq 2 −→ Rn(Pq) ←− Cn(∆) ∆ Tor1 (–,E) where

Cn(∆) = {M ∈ mod ∆ | dimM = (n, n, n − 1) and

M(Z−2) isomorphism ,M(Z−1) surjective }

−1 −1 Now put X = M(X−2)M(Z−2) and Y = M(Y−2)M(Z−2) . Thus X, Y ∈ End(M−1). Using the relations on the quiver ∆ it follows that   M(Z−1)M(X−2) = M(X−1)M(Z−2) M(Z−1)M(Y−2) = M(Y−1)M(Z−2)  M(Y−1)M(X−2) − M(X−1)M(Y−2) = M(Z−1)M(Z−2) from which one easily deduces

M(Z−1)(YX − XY − I) = 0 thus rank([Y, X] − I) = 1. It is clear that this procedure is reversible and this deﬁnes an equivalence (for n > 0) ∼ Cn = {(X, Y) ∈ Mn × Mn | rank([Y, X] − I) = 1}

Taking Gln-orbits one obtains the n-th Calogero-Moser space Cn.

It is not hard to prove that there is an alternative description of Cn, from which we obtain an analogous result as Theorem 5.1:

Cn(∆) = {M ∈ mod ∆ | dimM = (n, n, n − 1) and 1 Hom∆(M, p) = 0, Hom∆(p, M) = 0 for all p ∈ P }.

Let us now return to Sklyanin algebras. Although the category Cn(∆) has a fairly elementary description in Theorem 5.1 it is not so easy to handle. One may ask if one can simplify the description of Cn(∆) in the Sklyanin case as done in the Weyl algebra situation. At this point we mention the insight of Le Bruyn [17] that the representations M ∈ Cn(∆) in the Weyl case are actually determined by the three most left maps. We will try to mimic this idea.

37 5.6 Induced Kronecker quiver representations Let A be a quadratic three dimensional Artin-Schelter regular algebra.

Let ∆0 be the full subquiver of ∆ consisting of the vertices −2, −1 and let Res : Mod ∆ → Mod ∆0 be the obvious restriction functor. Res has a left adjoint which we denote by Ind. If e is the sum of the vertices of ∆0 then Ind = − ⊗k∆0 ek∆. Note that Res ◦ Ind = id. If M ∈ Mod ∆ we will denote Res M by M 0.

In general, we say that two objects A, B in an abelian category C are or- 1 thogonal (A ⊥ B) if HomC(A, B) = ExtC(A, B) = 0. For an object B ∈ Cf where Cf denotes the full subcategory of C consisting of the noetherian objects, ⊥ deﬁne B as the full subcategory of Cf which objects are

⊥ B = {A ∈ Cf | A ⊥ B}

By an argument of Bear [7], the functors Res and Ind deﬁne inverse equivalences

Res ⊥ −→ 0 mod ∆ ⊃ S0 ←− mod ∆ Ind

⊥ In particular, if M ∈ S0 then M = Ind Res M. This means that M is totally determined by Res M.

The following was already observed by Le Bruyn in the case of the homogenized Weyl algebra.

Lemma 5.3. Let A be a quadratic three dimensional Artin-Schelter regular 1 ⊥ algebra. If M ∈ Rn, n > 0 and M = Ext 2 (E, M) then M ∈ S0. Pq

Proof. We have RHom 2 (E, M) = M[−1], RHom 2 (E, O) = S . Thus Pq Pq 0

i i i+1 Ext 2 (M, O) = Ext (M[−1],S0) = Ext (M,S0) Pq ∆ ∆

In particular Hom∆(M,S0) = 0 and

1 2 2 ∗ Ext (M,S ) = Hom 2 (M, O) =∼ H ( , M(−3)) = 0 ∆ 0 Pq Pq where we have used Serre duality and Theorem 3.7. In particular the previous lemma translates into

⊥ Cn(∆) ⊂ S0 for n > 0.

38 5.7 Semistable representations Let us ﬁrst recall the notion of a (semi)stable representation. In general, let Q be a quiver without oriented cycles and write Q0, Q1 for respectively the set of vertices and edges of Q. Let θ ∈ ZQ0 be a dimension vector. A representation F of Q is called θ-semistable (resp. stable) if θ · dimF = 0 and θ · dimN ≥ 0 (resp. > 0) for every non-trivial subrepresentation N of F . Here we denote “·” Q0 P for the standard scalar product on Z :(αv)v · (βv)v = v αvβv.

It is a fundamental fact [23] that F is semistable for some θ if and only there exists G ∈ mod(Q) such that F ⊥ G. The relation between θ and dimG is such that the forms − · θ and χ(−, dimG) are proportional. Q0 Fix a dimension vector α ∈ Z and let Repα(Q) be the corresponding representation space, i.e. Y Repα(Q) = Mαh(i)×αt(i) (k)

i∈Q1 where the maps h, t : Q1 → Q0 associate to an arrow its begin and end vertex. It is a fundamental result that the isomorphism class of representations of dimension vector α are in one-one correspondence with the orbits of the group Gl(α) = Q Gl (k) acting on Rep (Q) by conjugation. v∈Q0 αv α

Associated to G ∈ mod(Q) there is a semi-invariant function φG on Repα(Q) such that {F ∈ Repα(Q) | F ⊥ G} = {φG 6= 0} (25) In particular (25) is aﬃne.

Let A be a quadratic three dimensional Artin-Schelter regular algebra. 1 We have seen that if M ∈ Rn, M = Ext 2 (E, M) then M = Ind Res M, which Pq means that M is completely determined by its restriction M 0 = Res M ∈ mod ∆0. In case A = H is the homogenized Weyl algebra, we furthermore 0 have that M (Z−2) is an isomorphism (see §5.5). For this insight we have to 2 consider line objects on Pq.

2 It is easy to see that for a line object S on Pq we have S(−1) ∈ X1, S = 1 ⊥ Ext 2 (E, S(−1)) ∈ S0 and dimS = (2, 1, 0). We have the following proposition Pq Proposition 5.4. Let A be a quadratic three dimensional Artin-Schelter regular 2 algebra. Let S = π(A/uA) be a line object on Pq where u = αx + βy + γz ∈ 1 2 A1. Write S = Ext 2 (E, S(−1)) ∈ mod ∆. Let M ∈ Rn( ) and write M = Pq Pq 1 Ext 2 (E, M) ∈ mod ∆. Then the following are equivalent: Pq 1. M 0 ⊥ S0 (where M 0 = Res M,S0 = Res S)

0 0 2. Hom∆0 (M ,S ) = 0

39 3. Hom 2 (M, S(−1)) = 0 Pq 4. M ⊥ S(−1) 5. The following linear map is an isomorphism

0 0 0 αM (X−2) + βM (Y−2) + γM (Z−2): M−2 → M−1

0 0 0 0 Proof. Equivalence of (1) and (2): M ⊥ S implies Hom∆0 (M ,S ) = 0. Con- 0 0 0 0 versely, if Hom∆0 (M ,S ) = 0 then by computing χ(M ,S ) = 0 we also have 1 0 0 Ext∆0 (M ,S ) = 0.

Equivalence of (2) and (3): By

0 0 0 Hom∆0 (M ,S ) = Hom∆0 (M , Res S) 0 = Hom∆(Ind M ,S) = Hom∆(M,S) 0 0 = H (RHom (M,S)) =∼ H (RHom 2 (M, S(−1))) ∆ Pq

= Hom 2 (M, S(−1)) Pq Equivalence of (3) and (4): We have [M] = [O] − n[P] and [S(−1)] = [S] − [P]. An easy computation shows χ(M, S(−1)) = 0. And by Serre duality

2 ∼ ∗ Ext 2 (M, S(−1)) = Hom 2 (S(2), M) = 0 Pq Pq since M is reﬂexive hence torsion free. We conclude that M ⊥ S(−1) if and only if Hom 2 (M, S(−1)) = 0. Pq

Equivalence of (4) and (5): Applying Hom 2 (, M) on the resolution of S(2) Pq 0 → O(1) → O(2) → S(2) → 0 gives

1 f 2 0 → Ext 2 (S(2), M) → M−2 −→ M−1 → Ext 2 (S(2), M) → 0 Pq Pq where we have used Theorem 3.7. It is clear that the map f is given by 0 0 0 αM (X−2) + βM (Y−2) + γM (Z−2). Thus f is an isomorphism if and only if 1 2 Ext 2 (S(2), M) = 0 and Ext 2 (S(2), M) = 0. Again using Serre duality this is Pq Pq equivalent with M ⊥ S(−1). In case A = H is the homogenized Weyl algebra we immediately ﬁnd that 0 M (Z−2) is an isomorphism. Indeed, let S = π(H/zH) then

∗ RHom 2 (M, S(−1)) = RHom 2 (M, i O 1 (−1)) =∼ RHom 1 (Li M, O 1 (−1)) Pq Pq ∗ P P P ∗ 1 and since Li M = OP

Hom 2 (M, S(−1)) =∼ Hom 1 (O 1 , O 1 (−1)) = 0 Pq P P P

40 2 In particular, the representations in Cn are θ-semistable for some θ ∈ Z . To see that θ = (−1, 1), compute that χ(−, dimS0) = − · (−1, 1).

Now assume that A = Skl3(a, b, c) is a three dimensional Sklyanin algebra. 2 2 Then one might also try to find a line object S on Pq such that HomP (M, S(−1)) 2 is zero for all M ∈ Rn(Pq). We did not manage to find such a (general) line object, however we were able to prove that for a fixed normalized line 2 bundle M ∈ Rn(Pq) there is a line object (which depends on M) such that 2 HomP (M, S(−1)) 6= 0. In particular it follows that all representations in Cn are semistable.

However, there is another interpretation. In case of the Weyl algebra we 1 ∗ 1 considered the structure sheaf OP of Ered = P . We then observed that i M = 0 1 OP which translated into M ⊥ S. Now for the Sklyanin case, consider the structure sheaf OE on the smooth elliptic 2 2 curve E. Let M ∈ Rn(Pq) be a normalized line bundle on Pq. Assuming that n > 0 we ﬁnd ∗ 1 ∗ HomE(i M, OE) = ExtE(i M, OE) = 0 In fact, since E is hereditary, we have by adjointness

∗ RHom 2 (M, i O ) =∼ RHom (i M, O ) = 0 Pq ∗ E E E

It is easy to see that i O = π(A/gA). But unfortunately, RHom 2 (E, i O ) is ∗ E Pq ∗ E not concentrated in a single degree. So the image of i∗OE under the generalized Beilinson equivalence cannot be identiﬁed with an object in mod ∆. In order to see this, again by adjointness we have

∗ RHom 2 (E, i O ) =∼ RHom (Li E, O ) Pq ∗ E E E and an easy veriﬁcation shows that

∗ 2 Li E = σ∗(L) ⊗ σ∗(L) ⊕ σ∗L ⊕ OE ∈ coh E Indeed:

Li∗E = i∗O(2) ⊕ i∗O(1) ⊕ i∗O

= B(2)e⊕ B(1)e⊕ Be so it is suﬃcient to show that

2 (Γ∗(OE))≥0 = B, (Γ∗(σ∗L))≥0 = B(1)≥0, Γ∗(σ∗(L) ⊗ σ∗(L)) ≥0 = B(2)≥0. The ﬁrst equality is by deﬁnition of B, while for example

0 0 σ−1 (Γ∗(σ∗L))≥0 = ⊕n≥0H (E, σ∗L ⊗ Ln) = ⊕n≥0H (E, L ⊗ Ln) 0 σ−1 0 0 = ⊕n≥0H (E, Ln+1) = ⊕n≥0H (E, σ∗Ln+1) = ⊕n≥0H (E, Ln+1)

= ⊕n≥0Bn+1 = B(1)≥0

41 2 Since E is hereditary we immediately have Ext 2 (E, i∗OE) = 0. And for example Pq from the exact sequence

·g 0 → A(−3) −→ A → A/gA → 0 one computes l ... −2 −1 0 ... 0 2 H (P , i∗OE(l)) ... 0 0 1 ... 1 2 H (P , i∗OE(l)) ... 6 3 0 ... 2 2 H (P , i∗OE(l)) ... 0 0 0 ... However, if we pick a degree zero line bundle U on E which is not of the form ∼ 0 2 O((o)−(3nξ)) for n ∈ N (In particular U 6= OE) then we do have H (Pq, i∗U) = 0. One obtains that RHom 2 (E, i U) is concentrated in degree one, say U[−1]. Pq ∗ We have dim U = (6, 3, 0) and putting V = Res U the following lemma is proved Lemma 5.5. Let A be a three dimensional Sklyanin algebra where σ has inﬁnite order. Consider a representation V ∈ mod ∆0 as above. Then

0 1. for all M ∈ Cn(∆) we have M ⊥ V , and

2. if p ∈ E then RHom∆0 (Res p, V ) 6= 0. 0 In particular, we (again) obtain that if M ∈ Cn(∆) then M is θ-semistable for θ = (−1, 1). The advantage now is that the representation V does not depend on M.

2 5.8 Second description of Rn(Pq)

Let A = Skl3(a, b, c) be a three dimensional Sklyanin algebra where σ has inﬁ- nite order.

0 We have shown that, given an object M in Cn(∆), then the restriction M = Res M of M to the Kronecker quiver ∆0 satisﬁes • dimM 0 = (n, n) = α • dim(Ind M 0) = n − 1 • M 0 ⊥ V and since χ(M 0,V ) = 0 we have

0 0 0 0 M ∈ Dn(∆ ) = {F ∈ Repα(∆ ) | dim(Ind M ) = n − 1 and Hom∆0 (F,V ) = 0} 0 ⊂ {F ∈ Repα(∆ ) | Hom∆0 (F,V ) = 0} = {φV 6= 0} 0 ⊂ {F ∈ Repα(∆ ) | F is semistable }

0 Now Dn(∆ ) a subset of a closed subset of an open subset of an aﬃne space. This is because the condition dim(Ind M 0) = n − 1 is a closed condition while

42 0 Hom(F,V ) = 0 is an open condition. Thus Dn(∆ ) is an aﬃne space. And 2 actually, we can show that the above properties characterise the image of Cn(Pq) under Res:

Theorem 5.6. Let A = Skl3(a, b, c) be a three dimensional Sklyanin algebra where σ has infinite order. Then the functors Res and Ind define inverse equivalences between Cn(∆) and 0 0 Dn(∆ ). Furthermore, if F ∈ Dn(∆ ) then F is θ-stable. and we finaly come to the main theorem

Theorem 5.7. Let A = Skl3(a, b, c) be a three dimensional Sklyanin algebra where σ has infinite order. 0 The affine variety Dn = Dn(∆ )/ Gl(α) is smooth and connected of dimension 0 2 2n. The isomorphism classes in Dn(∆ ) (and hence in Cn(∆) and Rn(Pq) and Rn(A)) are in natural bijection with the points in Dn. 0 Sketch of the proof. As pointed out above, Dn(∆ ) is affine so the orbit space 0 Dn = Dn(∆ )// Gl(α) is also affine. By Theorem 5.6 all representations in 0 0 Dn(∆ ) are θ-stable. This means that all Gl(α)-orbits on Dn(∆ ) are closed 0 and so Dn is really the orbit space for the Gl(α) action on Dn(∆ ). Therefore 0 we may write Dn = Dn(∆ )/ Gl(α). And this also proves that the isomorphism 0 classes in Dn(∆ ) are in natural bijection with the points in Dn.

0 To prove that Dn is smooth, it suﬃces to show that Dn(∆ ) is smooth. This follows for example by using the Luna slice theorem [19]. 0 To extend F ∈ Dn(∆ ) to a point in Cn(∆) we need to choose a basis in 0 (Ind F )0. Thus Cn(∆) is a principal Gln−1(k) ﬁber bundle over Dn(∆ ). In 0 particular Cn(∆) is smooth if and only Dn(∆ ) is smooth. Let x ∈ Cn(∆). Now x corresponds to some normalized line bundle M on 2 Pq and we have 1 1 Ext (x, x) = Ext 2 (M, M) ∆ Pq An easy computation shows χ(M, M) = χ(x, x) = 1 − 2n. We have Hom 2 (M, M) = k and by Serre duality Pq 2 ∼ ∗ Ext 2 (M, M) = Hom 2 (M, M(−3)) = 0 Pq Pq Thus 1 1 dimk Ext 2 (M, M) = 2n = dimk Ext (x, x) Pq ∆

This proves that Tx(Cn(∆)) is constant and hence Cn(∆) is smooth. Further dimension computations show that dim Dn = 2n.

Finally we say a few words how to prove the connectedness of Dn. To every point x ∈ Dn one may attach the Hilbert series hI (t) of a corresponding reﬂexive normalized rank one module I ∈ Rn(A). This induces a stratiﬁcation on Dn. Of course all strata have dimension ≤ 2n. We then show that there is precisely one statum which has maximal dimension 2n. This implies that Dn is connected.

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