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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 29th 2011

have created crystalline site

• CommentRowNumber2.
• CommentAuthorhilbertthm90
• CommentTimeMar 30th 2011

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?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 30th 2011

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

• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeMar 30th 2011

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 30th 2011

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.

• CommentRowNumber6.
• CommentAuthorzskoda
• CommentTimeMar 30th 2011

So what Grothendieck topology you take on $Aff/X_{dR}$ to have it equivalent as a site to the crystalline site ?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeMar 30th 2011

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?

• CommentRowNumber8.
• CommentAuthorzskoda
• CommentTimeMar 30th 2011
• (edited Mar 30th 2011)

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)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMar 30th 2011

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?

• CommentRowNumber10.
• CommentAuthorDavidRoberts
• CommentTimeMar 30th 2011
• (edited Mar 31st 2011)

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.

• CommentRowNumber11.
• CommentAuthorzskoda
• CommentTimeMar 31st 2011

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeMar 31st 2011

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.

• CommentRowNumber13.
• CommentAuthorzskoda
• CommentTimeMar 31st 2011
Good, if you can translate! I am now in a rush as well, but will do that part eventually, so we will nail it down :)
1. Added the recent result of Matthias Hutzler on what the big infinitesimal topos classifies. More details will most likely be added by himself in the near future.

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeApr 3rd 2019

• CommentRowNumber16.
• CommentAuthorDavidRoberts
• CommentTimeApr 3rd 2019

@Ingo - what’s the state of the art as far as getting what the various usual toposes classify? Which ones are still outstanding?

• CommentRowNumber17.
• CommentAuthorIngoBlechschmidt
• CommentTimeApr 3rd 2019
• (edited Apr 3rd 2019)

@David: To the best of my knowledge, we know what the following toposes classify:

• little Zariski topos: local localizations
• big Zariski topos: local rings
• big étale topos: separably closed local rings
• big fppf topos: fppf-local rings (conjecturally that’s the same as algebraically closed local rings)
• big fpqc topos: this is the same as the fppf topos if the usual finiteness conditions are imposed (which we need for all of the toposes anyway if we want them to classify a reasonable theory)
• big surjective topos: algebraically closed geometric fields
• big infinitesimal topos: surjective ring homomorphisms with nilpotent kernel

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:

• the $\neg\neg$ subtopos of the big Zariski topos (conjecturally: algebraically closed geometric fields which are integral over the base)
• big ph topos (conjecturally: algebraically closed valuation rings)
• big crystalline topos (Matthias is working on it)
• big Nisnevich topos
• big smooth topos
• big syntomic topos
• big cl topos
• big rh topos

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.