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I am trying to learn formal schemes. I tried to read the section in Hartshorne but I don't get very far from there since things are not done quite explicitly enough, at least in my opinion. I cannot read French, so EGA is out of the question. I would really appreciate it if you could tell me a good introduction to this topic.

flag	asked Dec 2 2010 at 3:38 Brian 876●2●15

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A free draft version of the Illusie chapter BCnrd suggested is available at (cdsagenda5.ictp.it//…). – Evan Jenkins Dec 2 2010 at 4:04

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Read Illusie's exposition (in English!) of the important and awe-inspiring results from EGA III$_1$ in the book "FGA Explained". That should clear up everything (motivation, technique, ways to think about and work with them, etc.). Note that the deepest results are in the proper case (just like GAGA). By the way, practice the math French; well worth the effort, and needs virtually no knowledge of real French. – BCnrd Dec 2 2010 at 4:04

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Perhaps you may try EGA I new edition after all. You will have to struggle with the language for a week, but after that you'll realize that the sentences are of a few types and the vocabulary is relatively limited. The exposition there is complete and lucid, so the effort will pay off. – Leo Alonso Dec 2 2010 at 9:56

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this may seem nuts, but you probably can read french if you try. e.g. "theoreme" = theorem. "epreuve" = proof = "demonstration", "algebrique" = algebraic,.... you do need a few verbs, but it is well worth the effort to learn them. take my word for it, give it a few more minutes... e.g. start with serre's fac. – roy smith Dec 3 2010 at 5:44

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Isn't this topic a generalization of zariski's theory of holomorphic functions? maybe those old papers would convey the idea. – roy smith Dec 13 2010 at 2:03

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Harry Gindi · Answer 1 · 2010-12-02 03:57:52Z UTC

I really liked this paper by Neil Strickland. It's quite good, but it only really covers the affine case. Be forewarned that, as stated in the introduction, scheme is taken to mean affine scheme unless explicitly noted. A very nice thing about the approach of this paper is that it works without many/any finiteness assumptions (since these often cannot be assumed for algebro-topological or homotopy-theoretic applications).

Leo Alonso · Answer 2 · 2010-12-02 11:03:47Z UTC

If you are interested in the infinitesimal structure of formal schemes (with emphasis on non necessarily adic maps) I suggest to look at my papers with Jeremías and Pérez "Infinitesimal Lifting and Jacobi Criterion for Smoothness on Formal Schemes" (Comm. Alg.) and "Local structure theorems for smooth maps of formal schemes" (JPAA). Also for the deformation theory of formal schemes look at M. Pérez "Basic deformation theory of smooth formal schemes" J. Pure Appl. Algebra 212 (2008), pp. 2381–2388 (MR 2009h:14006). I apologize for this self-promotion...

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