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# Homology, Homotopy and Applications

## Volume 4 (2002)

### Number 2

### The Roos Festschrift volume

### Noncommutative deformations of modules

Pages: 357 – 396

DOI: http://dx.doi.org/10.4310/HHA.2002.v4.n2.a17

#### Author

#### 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.

#### 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