International Congress of Mathematicians

Noncommutative Projective Geometry 952. Historical backgroundWe begin with a historical introduction to the subject. It really started withthe work of Artin and Schelter [2] who attempted to classify the noneommutativeanalogues F of a polynomial ring in three variables (and therefore of P 2 ). The firstproblem is one of definition. A "noneommutative polynomial ring" should obviouslybe a eg ring of finite global dimension, but this is too general, since it includes thefree algebra. One can circumvent this problem by requiring that dim/. F, growspolynomially, but this still does not exclude unpleasant rings like k{x,y}/(xy) thathas global dimension two but is neither noetherian nor a domain. The solution isto impose a Gorenstein condition and this leads to the following definition:Definition 1 A eg algebra R is called AS-regular of dimension d if gl dim R = d,GKdim F < oo and R is AS-Gorenstein; that is, Ext*(fc,F) = 0 for i ^ d butExt rf (fc, R) = k, up to a shift of degree.One advantage with the Gorenstein hypothesis, for AS-regular rings of dimension3, is that the projective resolution of k is forced to be of the form0 —•+ F —• R (n) —• R (n) —• R —•+ jfc —•+ 0for some n and, as Artin and Schelter show in [2], this gives strong informationon the Hilbert series and hence the defining relations of R. In the process theyconstructed one class of algebras that they were unable to analyse:Example 2 The three-dimensional Sklyanin algebra is the algebraSkl 3 = Skl 3 (a, 6,c) = k{xo,xi,a; 2 }/'(aXjXj+i + bxj+ix» + cx 2 +2 : i £ Z 3 ),where (a, 6, c) £ P 2 \ F, for a (known) set F.The original Sklyanin algebra Skl 4 is a 4-dimensional analogue of Skl 3 discoveredin [23]. Independently of [2], Odesskii and Feigin [18] constructed analogues ofSkl 4 in all dimensions and coined the name Sklyanin algebra. See [13] for applicationsof Sklyanin algebras to another version of noneommutative geometry.In retrospect the reason Skl 3 is hard to analyse is because it depends upon anelliptic curve and so a more geometric approach is required. This approach camein [6] and depended upon the following simple idea. Assume that F is a eg algebrathat is generated in degree one. Define a point module to be a cyclic graded (right)F-module M = @ i>0 M t such that dim?. M t = 1 for all i > 0. The notation isjustified by the fact that, if F were commutative, then such a point module Mwould be isomorphic to k[x] and hence equal to the homogeneous coordinate ring ofa point in Proj R. Point modules are easy to analyse geometrically and this providesan avenue for using geometry in the study of eg rings.We will illustrate this approach for S = Skl 3 . Given a point module M =(J) Mi write Mi = m,fc for some m, £ M t and suppose that the module structureis defined by m»Xj = Ayro, + i for some Ay £ k. If / = Yl fìj' x ì' x j ' 1S > one °f t nerelations for S, then necessarily mo/ = (^/yAojAij)m 2 , whence ^fij^oi^ij = 0.