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Where can I found a description of the deformation theory for modules?Is it possible to deform a free module in such way that each fibre of the deformation is still free?
A free module is rigid, because $Ext^1(E, E)=0$ for any free module $E.$
For deformation theory see for example ``Functors of Artin Rings''
by Michael Schlessinger.
Required, but never shown