On ideal classes of three dimensional Sklyanin algebras 4 expos´esau seminaire d'alg`ebre Universit´eJean 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 ) = fV 2 mod A0 j V cyclic and dimk V = ng= iso (1) 2 For V 2 Hilbn(A ) its annihilator AnnA0 (V ) = fa 2 A0 j a · V = 0g is an ideal of A0 of finite codimension, and this correspondence is reversible: 2 Hilbn(A ) = fI ⊂ A0 ideal j dimk A0=I = ng= 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 f.g. torsion free rank one A0-modules g= 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 2 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 2 V for which v · A0 = V . Thus 2 n [X; Y] = 0 V 2 Hilbn(A ) 7! data X; Y 2 Mn(k); v 2 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 2 Hilbn(A ) 7! data X; Y 2 Mn(k); v 2 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, x2.2]. 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) rr Writing rep(n;1) Q for the representations of the quiver Q with dimension vector (n; 1) we find 2 ∼ Hilbn(A ) = f(X; Y; i; j) 2 rep(n;1) Q j [X; Y] = ij n and khX; Yi · i(1) = k g= Gln(k) (5) where the group Gln(k) acts by conjugation −1 −1 −1 8g 2 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 di- mensional right A1-modules fV 2 mod A1 j V cyclic and dimk V = ng 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, fI ⊂ A1 right ideal j dimk A1=I = ng = ; 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) = f finitely generated torsion free rank one A1-modules g= 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 f.g. torsion free rank one A1-modules g= iso n where the variety Cn is the so-called nth Calogero-Moser space Cn = f(X; Y; i; j) 2 rep(n;1)(Q) j [X; Y] + I = ijg= 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 commu- tative 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 = f(X; Y) 2 Mn(k) j rank([Y; X] − I) ≤ 1g= Gln(k) where Gln(k) acts by simultaneous conjugation.