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infinitesimal 'spaces' is a serious issue in noncommutative (and commutative) geometry: they serve as a base of a Grothendieck-Berthelot crystalline theory and are of big importance for the D-module theory.

Can anybody point out the standard reference for this topic? I tried to look for it at nLab, but it seems it did not tell the reference in the language of algebraic geometry.

I am not very familiar with French,so the English manuscript is better, however, French one is fine.

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  • $\begingroup$ Maybe Berthelot-Ogus "Notes on Crystalline Cohomology"? $\endgroup$
    – Kevin H. Lin
    Commented Jan 14, 2010 at 9:25
  • $\begingroup$ Isn't the whole thing about noncommutative geometry that there is no presentation of a scheme as a topological space with a sheaf of rings, but only as a functor of points in the smooth grothendieck topology on the category of nonabelian affine schemes (which is the opposite category of an appropriate category of rings)? My point here is kinda that you might have some trouble finding any stuff at all in noncommutative geometry written in commutative AG language (i.e. not written as functors of points.) I may have misunderstood what you meant though. $\endgroup$
    – Harry Gindi
    Commented Jan 14, 2010 at 12:18
  • $\begingroup$ It sounds like you're looking for references on commutative Artinian local rings, since their spectra are the "fat points" in algebraic geometry. Alternatively, you may want to look for references on deformation theory, since they use such structures a lot. Illusie's Complexe Cotangent et Deformations seems to be the standard reference, but Hartshorne also wrote a book relatively recently. $\endgroup$
    – S. Carnahan ♦
    Commented Jan 14, 2010 at 19:11
  • $\begingroup$ I'm afraid I don't understand what topic you're seeking a reference in. What sort of "spaces" are you asking about? Vector spaces? Topological spaces? Schemes? Algebraic spaces? Sheaves on some site? What is the "serious issue" you're referring to? $\endgroup$
    – Clark Barwick
    Commented Jan 20, 2010 at 0:45
  • $\begingroup$ If I understand your question correctly, for the commutative case you should have a look at EGA IV, §16 and any book discussing formal completion along subschemes, say Hartshorne or EGA I. As for the connection of D-Modules and Crystals, the already mentioned book of Berthelot-Ogus is very nice. Then there is also Berthelot's LNM Text on Crystalline Cohomology. I also really enjoyed these notes: math.harvard.edu/~gaitsgde/grad%5F2009/SeminarNotes/… $\endgroup$
    – Lars
    Commented Jan 21, 2010 at 21:34

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I think that the best reference for the infinitesimal site, especially if one is motivated to learn crystalline cohomology, is found in Grothendieck's lectures "Crystals and the De Rham Cohomology of Schemes", notes by Coates and Jussila. This appears, in English, as number 9 of 10 in the famous "Dix Exposes". It is an absolutely beautiful, and very readable article. The entire Dix Exposes can be obtained freely online, as a PDF file, from Leila Schnepps's Grothendieck Circle website.

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