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have created crystalline site
Do we have to worry about positive characteristic? I don’t know anything about the crystalline site, but I’ve been hunting for readable references for awhile to learn. Usually people take 50 pages or so and go through all sorts of divided power structures and things that I’ve never sat down and figured out what they are talking about. I’m skeptical of such a compact, simple definition. Maybe all this stuff comes out if you write out what it actually means to be a coequalizer of the two projections out of the formal neighborhood of the diagonal?
Let me see, maybe I am missing something, I just meant to record the statement as stated on p. 7 of Simpson-Teleman de Rham theorem for $\infty$-stacks, where it says
Thus, the over category $Aff/X_{dR}$ — whose objects are, by definition, natural transformations from representable functors to $X_{dr}$ — agrees with Grothendieck’s (big) infinitesimal site of X of [Gr],
where [Gr] is Grothendieck, Crystals and the de Rham cohomology of schemes In: Dix Expos´es sur la cohomologie des schémas, 306–358, North-Holland, Amsterdam 1968
Divided power structures are used, among other things, to describe Taylor series for functions in positive characteristic ($p$, say), because every term beyond $x^p$ is obviously not naively defined.
I think dividing out by the nilradical is a pretty substantial alteration, and that’s where the interesting stuff happens.
Right, so they assume characteristic 0 (complex or analytic site). I have edited the entry a bit. But clearly more should be said here eventually.
So what Grothendieck topology you take on $Aff/X_{dR}$ to have it equivalent as a site to the crystalline site ?
According to the Simpson-Teleman piece that I mentioned, you take the big site-topology.
But can you help me with the terminology: when do we say “infinitesimal site” and when “crystalline site”?
Is it right that in char = 0 both notions coincide?
Look I know how to induce the topology on the overcategory, but which topology do you use in the first place on Aff for this reason; there exist Zariski big site, fpqc big site and so on. So what is the topology which will agree with crystalline after the translation ? Flat ? Zariski ? (my guess would be Zariski)
As I said, I am just following the statement in Simpson-Teleman, page 7. They state this for Zariski and étale topology.
But apparently we need better sources. Does anyone have an electronic copy of the original Grothendieck article or else another good account?
Only a paper copy of ’Dix exposes…’, I’m afraid. I could dig it out and have a look.
Edit: and I don’t have the relevant section … :( Sorry.
Dix exposes is online at many places, huge file. But it will not answer the question, as it does not have the de Rham space approach I suppose, but the original crystalline site. For the original crystalline site good sources are many online available papers by Ogus and by (his former student) Martin Olsson.
But it will not answer the question, as it does not have the de Rham space approach
But when I see the definition I can translate it to the de Rham space.
I mean, the de Rham space is a very trivial thing: the presheaf defined by $X_{dR} : Spec R \mapsto X(Spec R_{red})$. This is just a sheaf-theoretic way of saying what is said in the definition of infinitesimal site anyway.
All I need to see is what the definition of crystalline ssite boils down to in char 0 and what the terminiology convention is concerning “infinitesimal site” and “crystalline site”. I guess the latter is just the adjustment of the former with those $W_n$-quotients invoked to make things work.
I don’t have time to search for this right now. Will do it later. Unless somebody else does.
@Ingo - what’s the state of the art as far as getting what the various usual toposes classify? Which ones are still outstanding?
@David: To the best of my knowledge, we know what the following toposes classify:
Some of the “little” variants are missing from this list, but could be rather easy to obtain or might already be folklore. The Zariski case is folklore, the canonical reference being Moerdijk and Mac Lane’s Sheaves in Geometry and Logic. The étale case is in Monique Hakim’s thesis and Gavin Wraith’s Generic Galois theory of local rings. Fppf, fpqc and surjective are in Section 21 of these notes. Infinitesimal is in Matthias’s master thesis.
We don’t know (at least there is no written account of) what any of the remaining toposes classify, in particular:
Some ramblings about the two conjectures is towards the end of above-cited Section 21. For many of these toposes, we at least know what their points are.
Note that today we have much better technology at our disposal to tackle these questions: I’m thinking of Diaconescu’s theorem, the coherent account in Olivia’s book, composition of internal geometric theories and the recognition of the notion of compact objects as being important in this context. (Matthias deduces his characterization more or less as a corollary of what the compact objects in a certain category are.) These help to strip away technicalities, allowing us to focus on the mathematical contents.
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