Noncommutative algebraic geometry

Abstract

The need for a noncommutative algebraic geometry is apparent in classical invariant and moduli theory. It is, in general, impossible to find commuting parameters parametrizing all orbits of a Lie group acting on a scheme. When one orbit is contained in the closure of another, the orbit space cannot, in a natural way, be given a scheme structure. In this paper we shall show that one may overcome these difficulties by introducing a noncommutative algebraic geometry, where affine "schemes" are modeled on associative algebras. The points of such an affine scheme are the simple modules of the algebra, and the local structure of the scheme at a finite family of points, is expressed in terms of a noncommutative deformation theory proposed by the author in \cite{Laudal2002}. More generally, the geometry in the theory is represented by a {\it swarm}, i.e. a diagram (finite or infinite) of objects (and if one wants, arrows) in a given $k$-linear Abelian category ($k$ a field), satisfying some reasonable conditions. The noncommutative deformation theory refered to above, permits the construction of a presheaf of associative $k$-algebras, locally {\it parametrizing} the diagram. It is shown that this theory, in a natural way, generalizes the classical scheme theory. Moreover it provides a promising framework for treating problems of invariant theory and moduli problems. In particular it is shown that many moduli spaces in classical algebraic geometry are commutativizations of noncommutative schemes containing additional information.