Consider M a locally free $\mathcal{O}_{\mathbb{C}^{n}}$-module. does exist a theory of deformation for that type of object? I would like to know, which conditions has to satisfy the total space of a one (but also higher) parameter deformation of M in such way that each fiber is locally free.

  • $\begingroup$ Would you mind clarifying what definition of "deformation" you are using? (In my world, the Hecke algebra is a very interesting deformation of the group algebra of a complex reflection group which is nonetheless "trivial" according to the "infinitesimal" point of view). $\endgroup$ – GS May 10 '10 at 14:51
  • $\begingroup$ What I have in mind is the following type of situation:consider $\mathbb{C}^{n+1}$ with variables $x_1, \dots, x_n,t$ and X an $\mathcal{O}_{\mathbb{C}^{n+1}}$-module such that when t=0 I get M, I would like to know what I sould expact from X if for each fixed t, X restrict to a locally free module.I mention deformations because, I'm deforming a geometrical object and I can attach to it the module M, and so I think that deforming my object I'm also deforming M in some sense...Is it more clear now? $\endgroup$ – Michele Torielli May 10 '10 at 15:02
  • $\begingroup$ Yes, thanks! You're really thinking about "global" deformations. I'm not an expert, just wanted to understand the question better. $\endgroup$ – GS May 10 '10 at 15:19

I think the theory you are looking for can be found in Th. 8.5.3, chap. 8 (p. 210) of the book by B. Fantechi et al., "FGA explained".

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