Luận án tiến sĩ: Ideals of three dimensional Artin-Schelter regular algebras

DOCTORAATSPROEFSCHRIFT 2006 | Faculteit Wetenschappen Ideals of three dimensional Artin-Schelter regular algebras Bijgevoegde stelling voorgelegd tot het behalen van de graad van Doctor in de Wetenschappen, richting wiskunde, te verdedigen door: Koen DE NAEGHEL Promotor: Prof. VAN DEN BERGH universitel D/2006/2451/1 „»hna Ssel† UMI Number: 3204340 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.

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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 Contents Acknowledgements Vv Introduction vi 1 Preliminaries and basic tools 1 1.11 Grothendieck group and Euler form. LH HQ HQ.2 Algebras and modules .4 Graded algebras and modules. Q Q Q LH ng HQ ngu Quà vat 6 1.8 Artin-Schelter regular algebras .9 Three dimensional Artin-Schelter algebras.

000000000 cc eee eee 11 1.2 Dimension, multiplicity and Hilbert series .11 Filtered algebras and modules. 28 2 Ideals of quadratic Artin-Schelter algebras 31 2.1 Motivation, main results and analogy.1 Hilbert schemes on affine planes .2 Hilbert schemes on projectiveplanes.3 Stable vectorbundles on the projective plane .2 From reflexive ideals to normalized line bundles bundles .1 Torsion free and reflexive objects.2 The Grothendieck group and the Euler form for quantum planes 44 2.3 Normalized rank one objects 2. ee es 44 i ii CONTENTS 2.4 Cohomology of line bundles on quantum planes. Restriction of line bundles to the divisorŒ.4 From line bundles to quiver representations .1 Generalized Beilinson equivalence.2 Point and line representations.3 First description of Rp(P2) .4 Line bundles on P2 with invariantone.5 Induced Kronecker quiver representations .7 Second description of Ry, (P2) and proof of Theorem1 .8 Description of the varieties D,, for Sklyanin algebras and proof of Theorem 2 .9 Filtrations of line bundles and proof of Theorem3.

71 3 Hilbert series of ideals of quadratic Artin-Schelter algebras 73 3.1 Introduction and main results.2 Notations and conventions.0000 ce ee 22 eee 78 3.2 Proof that the conditions in Theorem 6 are necessary .3 Proof that the conditions in Theorem 6 are sufficient .0 0000 eee eee ee eee 82 3.4 Proof of other properties of Hilbert series .5 The stratification by Hilbert series. ee ee ee 90 3. cee ee ee 93 4 Modules of GK-dimension one over quadratic Artin-Schelter alge- bras 95 4.4 Proof of Theorem 7. eee ee ee 102 5 Incidence between strata on the Hilbert scheme of points on the projective plane 103 5.1 Introduction and main result .2 Outline of the proof of the main theorem .2 Four sets of conditions .1 Translation of the length zero condition .2 Translation of the dimension condition.3 Translation of the tangent condition.

ee eee eee eee 114 5.41 Truncated point modules .2 A complex whose homology isJ .8 The Hilbert scheme ofanideal.4 Estimating the dimension of Ext)(J,J) .5 Estimating the dimension of Ext) (J 0 121 5.6 Tying things together. LH HQ es 123 6 Ideals of cubic Artin-Schelter algebras 127 6.1 Introduction and main results.2 From reflexive ideals to normalized line bundles.1 Torsion free and reflexive objects .2 The Grothendieck group and the Euler form for quantum quadrics132 6.3 Normalized rank one objects .4 Cohomology of line bundles on quantum quadrics.3 Hilbert series of ideals and proof of Theorem lä.4 Ideals of linear cubic Artin-Schelter algebras.2 Hilbert scheme of points.5 Some results on line and conic objects. ee eee ee ee ee 144 6.6 Restriction of line bundles to the divisorC .7 From line bundles to quiver representations .1 Generalized Beilinson equivalence.2 Point, line and conic representations .3 First description of Rin.4 Line bundles on X with invariants (1,0) and (1,1) .5 Description of Rin.n,)(X) for the enveloping algebra .6 Restriction to a full subquiver.8 Second description of Rin.9 Descriptions of the varieties („„„„) for the enveloping algebra 164 6.10 Description of the varieties Din, n,) for generic type A.8 Eiltrations of line bundles and proof of Theorem12.9 Invariant ring of the first Weyl algebra and proof of Theorem 14. 169 A Serre duality for graded rings 171 iv CONTENTS B Upper semi-continuity for noncommutative Proj 175 C Hilbert series of ideals with small invariants 177 C.1 Quadratic Artin-Schelter algebras.2 Cubic Artin-Schelter algebras.

Q Q Q ee 179 D A visual criterion for incidence problems of length zero 181 E Maple programs 183 E.1 Procedures for Chapter 3: Examples .2 Procedures for Chapter 5: Examples .3 Procedures for Chapter3. eee ee ee 193 E.4 Additional procedures for Chapter5. 200 F Hilbert graphs and combinatorics 211 F.1 Hilbert graphs and incidence problems for low invariants.2 General Hilbert graphs. ee ee ee t 215 F.

eee 219 G An inequality on broken chessboards 223 G. Q Q n kg gà gà gà va 223 G. LH nu ng cv kg k kg k kia 226 G.1 Partitions and chess Ferrers graphs.2 From partitions to Castelnuovo functions .4 Proof that the condition in Theorem A is necessary .5 Proof that the condition in Theorem A is sufficient. eee ee ee ee es 233 G.

ee ka và 234 Bibliography 237 Acknowledgements Firstly, there are many people I would like to thank for a variety of professional reasons. I would like to gratefully acknowledge the enthusiastic supervision of Michel Van den Bergh during this work. He did a perfect job in guiding me through the mathe matical landscape, and I would have been lost without him. Thank-you to the members of the jury, for managing to read the whole thing thoroughly.

I also state my gratitude to 5. Paul Smith and Nicolas Marconnet for sharing mathematical ideas with me. Thanks are due also to several researchers at other organisations for giving me all kinds of possibilities and for their invitations: Jacques Alev, Roland Berger, Bernhard Keller, S. Paul Smith, Freddy Van Oystaeyen.

] am grateful for the predoc position funded by the European Research Training Network Liegrits. Colleagues and professors at Hasselt, University are thanked for their care and attention. Secondly, as I have always tried to separate work and personal life the latter de- serves special attention. It is difficult to overstate my gratitude to friends and special friends, in past and present time, wherever they are.

Their influence on me - and therefore also on my thesis - has been immeasurable. However I do not think this is the correct place to list them, moreover it is far to delicate to write them all down. They will be thanked personally. Lastly, and most importantly, I am forever indebted to my parents.

They raised me, supported me, taught me, and loved me. Above all things they had faith in me. To them I dedicate this thesis. Introduction In 1927 Heisenberg discovered that uncertainties, or imprecisions, always turned up if one tried to measure the position and the momentum of a particle at the same time.

His 14-page letter to Pauli evolved into a published paper [42] in which Heisen- berg presented to the world what became known as the uncertainty principle. In order to study this principle, Weyl introduced the algebra A; = k(x, y)/(xy — yx — 1) as the k-algebra generated by position and momentum operators z, y. For simplicity k will be an algebraically closed field & of characteristic zero. The noncommutativity of the generators reflects the Heisenberg uncertainty principle.

The algebra A; is called the first Weyl algebra, it is the basic example of a non- commutative noetherian domain and has been studied in various papers and books. In particular A; is simple i. there are no non-trivial two-sided ideals. However there are plenty one-sided ideals of A; and it is a natural question to describe them.

In 1994 Cannings and Holland [22] classified right A-ideals, by means of the adelic Grassma- nian. A few years later Wilson [84] found a relation between the adelic Grassmanian and the Calogero-Moser spaces. It turned out [17] that the orbits of the natural action of the automorphism group Âut(4¡) on the set of right A,-ideals are indexed by the set of natural numbers N, and the orbit corresponding to n € Ñ is in natural bijection with the n-th Calogero-Moser space Ca = {(X, Y) € Mfa(k) x Ma(k) | rank(YX — XY — I) < 1}/ GI, (4), a connected smooth affine variety of dimension 2n [84]. On the other hand, in 1995 Le Bruyn [51] proposed an alternative classification method based on noncommutative algebraic geometry.

His idea was to consider the homogenized Wey] algebra H, the algebra obtained by adding a third variable z of degree one to A; which commutes with x,y, making the relation cy — yx — 1 homogeneous. Then A; is thought of as the coordinate ring of an open affine part of a noncommutative space P2 = Proj H in the sense of Artin and Zhang [10] ie. Proj H is the quotient of the abelian cate- gory of finitely generated graded right H-modules by the Serre subcategory of finite dimensional modules. The problem of describing Á¡-ideals then becomes equivalent to describing certain objects on the noncommutative projective plane Đệ.

They may be used to describe moduli spaces, just as in the ordinary commutative case. In 2002 this idea of Le Bruyn was picked up and worked out by Berest and Wilson [16] to prove directly the relation between 4i-ideals and Calogero-Moser spaces C,,. This work starts from the observation that the algebra H is a so-called three dimensional Artin-Schelter regular algebra, see Chapter 1 for these preliminary def- initions. This class of graded algebras was introduced by Artin and Schelter [5] in 1986 and classified a few years later by Artin, Tate and Van den Bergh (7, 8] and Stephenson [72, 73].

They are all noetherian domains of Gelfand-Kirillov dimen- sion three and may be considered as noncommutative analogues of the polynomial ring k[x,y,z]. To each of them there is an associated noncommutative projective surface Proj A. Let us further assume A is generated in degree one. It turns out vi INTRODUCTION vil [5] there are two possibilities for such an algebra A.

Hither there are three gener- ators z,y,z and three quadratic relations (we say A is quadratic) or two genera- tors x,y and two cubic relations (A is cubic). If A is quadratic then A is Koszul and has Hilbert series hA(#) = 1 + 3t + 6/2 + 10¢2+.--- = 1/(1 — £)!, and we may think of Pa = Proj A as a noncommutative projective plane. In case A is cubic then ha(t) = 1+2t+ 4t?+46t3+--.=1/(1—t)?(1—+?), we then write (P’ x P'), = ProjA which we think of as a noncommutative quadric surface. The generic class of quadratic and cubic Artin-Schelter algebras are usually called type A-algebras [5], in which case the relations are respectively given by ayz + bzy + ca? =0 aza + baz + cy’ 2 = 0 and { a2z az2?u + bụz + ary? + cx? = 0 + beyx + auz2 + cy? = 0 (1) ary + byx + cz? = 0 where a, b,c € k are generic scalars.

It was shown in [7| that a three dimensional Artin-Schelter regular algebra A generated in degree one is completely determined by a triple (E, ơ, 7) where either ej: E&P if A is quadratic, resp. j: E YP! xP if A is cubic; or e 7: E © IP? is an embedding of a divisor E of degree three if A is quadratic, resp. j: © P!