OK, so currently I am writing a bit of material on etale sites, points of etale sites and (upcoming) morphisms of etale sites. The reason for doing this is that I want to verify that my claim in the preceding post is correct.
Also, I finished rewriting the material on formally unramified, etale and smooth morphisms of schemes. I introduced the notion of a thickening of schemes (which is a closed immersion whose ideal is locally nilpotent), and a first order thickening. Then I introduced the notion of a universal first order thickening of a scheme formally unramified over another scheme. Using this a formally etale morphism is one which is formally unramified such that the universal first order thickening is trivial. I also rewrote some of the material on formally smooth morphisms, splitting the main result into separate lemmas (and slightly generalizing the result). I think this has substantially improved the exposition.
I want to use the same ideas to discuss formally unramified, etale and smooth morphisms of algebraic spaces, as well as modules of differentials and conormal sheaves for morphisms of algebraic spaces (see my previous post). In order to do this it is going to be really helpful to have the claim of the preceding post, so I am trying to finish this up first.
Meanwhile, work on the ideas mentioned in this post has been delayed a bit, but I hope to return to it soon. The reason for the delay is that I decided that the construction of the (badly named) f_!X mentioned there should go in the chapter “More on Morphisms of Spaces”. As a result I started thinking about some of the material that should go into this chapter, and so on and so forth.
PS: I am in the market for a good symbol to use as a replacement for f_!X…