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Is it true that for any deformation theory, the deformation constructed from a basis of $T^1$ is formally versal (let's assume that $T^1$ is finite dimensional)? Moreover, does anyone has a reference about the subject?

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 What do you mean by "the deformation constructed from a basis" of $T^1$? One can make plenty of non-versal infinitesimal deformations so that the "classifying map" from the formal deformation ring is an isomorphism on tangent spaces but not formally smooth. Have you tried thinking about examples, say even with 1-dimensional deformation ring (with 1-dimensional tangent space)? And reference on what subject? Versal deformations? $T^1$? Please be more specific in your question. – Boyarsky Jul 7 2010 at 13:35
sorry. I was not thinking of formal deformation theory(i.e. over artinian rings) but of deformation theory like deformation of singularities over complex spaces where $T^1$ is the quotient of e certain Hom. – Michele Torielli Jul 7 2010 at 13:46
Dear Michele: OK, then please rewrite the question a bit more fully so it is clear what you are asking (e.g., the use of the word "formally" in the title may merit clarification). It still feels like the answer will be "no", but you should clarify how a basis of $T^1$ gives rise to a specific deformation (of whatever sort you wish to consider). – Boyarsky Jul 7 2010 at 14:53

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