Alex Perry wrote a chapter on formal deformation theory for the stacks project following Schlessinger and Rim. Please read the introduction of that chapter for more information.

I intend to work on this chapter a little bit more in the near future in order to allow for finite residue field extensions (i.e., work with Λ —> k of finite type). The way the chapter is written however, I believe only minor changes will have to be made.

Once this is done we intend to use this material to study the formal local structure of algebraic stacks and to explain Artin’s criteria for Algebraic Stacks. One big obstruction looming in the future is the general Neron desingularization (Popescu). I’m not yet sure how to deal with this.

More immediately what we really need now is a couple of examples where the theory applies directly as written up. Alex and I listed a few obvious examples at the end of the chapter. If you feel like writing one of these up (should not be more than a few pages) using the framework we have in place please email me (so we don’t do double work).

Dear Johan,

I have some personal notes on Artin’s algebraization theorem. I would be happy to contribute those if they would be useful (although they are rather rough). The notes presuppose Artin approximation / Popescu’s generalization.

Best regards,

Jason

@Jason: Great! Send them over.

@Everybody: I’ve found it to be very useful to have any kind of rough version of a chapter to start with, because then we can make incremental changes to fit it better into the stacks project.

@Johan: I sent the notes, but they cover less than I remembered. Sorry!