Homology, Homotopy and Applications
Volume 4 (2002)
The Roos Festschrift volume
Noncommutative deformations of modules
Pages: 357 – 396
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.
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