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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?

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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.

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Well, this certainly depends. If $E$ is an $R$-module and you want to deform it as an $R$-module, then it's rigid. But often it happens that a module is free over some $S$ but you want to deform it over some other $R$. In that case the deformation problem is not rigid. By the way, I'm not so sure Schlessinger's paper handles the case of modules, but I may have a faulty recollection here. – Daniel Larsson May 1 '10 at 12:02
Oh, and do a MathScinet search of deformations of modules and you will likely get hundreds of hits. – Daniel Larsson May 1 '10 at 12:04
what about locally free? – Michele Torielli May 2 '10 at 7:21

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