ALGEBRA, GEOMETRY AND MATHEMATICAL PHYSICS BANACH CENTER PUBLICATIONS, VOLUME 93 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2011 GEOMETRY OF NONCOMMUTATIVE ALGEBRAS EIVIND ERIKSEN BI Norwegian Business School, N-0442 Oslo, Norway E-mail: eivind.eriksen@bi.no ARVID SIQVELAND Buskerud University College, P.O. Box 235, N-3603 Kongsberg, Norway E-mail: arvid.siqveland@hibu.no Abstract. There has been several attempts to generalize commutative algebraic geometry to the noncommutative situation. Localizations with good properties rarely exist for noncommutative algebras, and this makes a direct generalization difficult. Our point of view, following Laudal, is that the points of the noncommutative geometry should be represented as simple modules, and that noncommutative deformations should be used to obtain a suitable localization in the noncommutative situation. Let A be an algebra over an algebraically closed field k. If A is commutative and finitely generated over k, then any simple A-module has the form M = A/m, the residue field, for a maximal ideal m ⊆ A, and the commutative deformation functor Def M has formal moduli ˆ Am . In the general case, we may replace the A-module A/m with the simple A-module M, and use the formal moduli of the commutative deformation functor Def M as a replacement for the complete local ring ˆ Am . We recall the construction of the commutative scheme simp(A), with points in bijective correspondence with the simple A-modules of finite dimension over k, and with complete local ring at a point M isomorphic to the formal moduli of the corresponding simple module M. The scheme simp(A) has good properties, in particular when there are no infinitesimal rela- tions between different points, i.e. when Ext 1 A (M,M ′ )=0 for all pairs of non-isomorphic simple A-modules M,M ′ . It does not, however, characterize A. We use noncommutative deformation theory to define localizations, in general. We consider the quantum plane, given by A = k〈x, y〉/(xy − qyx), as an example. This is an Artin-Schelter algebra of dimension two. 2010 Mathematics Subject Classification : Primary 14A22; Secondary 16G20. Key words and phrases : noncommutative algebraic geometry, simple modules. The paper is in final form and no version of it will be published elsewhere. DOI: 10.4064/bc93-0-6 [69] c  Instytut Matematyczny PAN, 2011