Noncommutative deformations of modules

Abstract

The classical deformation theory for modules on a $k$-algebra, where $k$ is a field, is generalized. For any $k$-algebra, and for any finite family of $r$ modules, we consider a deformation functor defined on the category of Artinian $r$-pointed (not necessarily commutative) $k$-algebras, and prove that it has a prorepresenting hull, which can be computed using Massey-type products in the $Ext$-groups. This is first used to construct $k$-algebras with a preassigned set of simple modules, and to study the moduli space of iterated extensions of modules. In forthcoming papers we shall show that this noncommutative deformation theory is a useful tool in the study of k-algebras, and in establishing a noncommutative algebraic geometry.