Noncommutative deformations of modules

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.

Keywords

modules, deformations of modules, formal moduli, moduli of iterated extensions, Hochschild cohomology, Massey products, swarm of modules, algebra of observables, modular substratum, quivers

2010 Mathematics Subject Classification

16E30, 16G70

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Published 1 January 2002