Just this morning I finished revising the chapter on formal deformation theory that was written by Alex Perry. Next week I hope to put this chapter to use in the stacks project, and start writing about Artin’s criteria.
In particular, I hope to get back to what I said in this post. Put in another fashion, I want to prove an equivalence of the form “Artin’s criterion holds for stacks in groupoids over S” <=> “S is good” where good is a property of Noetherian base schemes to be determined. Goodness might be closely related to a condition like “completions of local rings are limits of flat finitely presented algebras”. I believe this could be fun!