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In his article "Crystals and the de Rham cohomology of schemes" in the collection "Dix exposes sur la cohomologie des schemas" Grothendieck defines the (small) infinitesimal site of an $S$-scheme $X$ using thickenings of usual opens. He then proceeds to prove that in characteristic $0$ the cohomology with coefficients in $\mathcal{O}_{X}$ computes the algebraic de Rham cohomology of the underlying scheme. This is remarkable, because the definition of the site does not use differential forms and it is not necessary for $X/S$ to be smooth.

This fails in positive characteristics, and as a remedy, Grothendieck sugessts adding the additional data of divided power structures to the site, which he then calls the "crystalline site of $X/S$". This site then has good cohomological behaviour (e.g. if $X$ is liftable to char. $0$, then cohomology computed with the crystalline topos is what it "should be"). The theory of the crystalline topos was of course worked out very successfully by Pierre Berthelot.

My question is: Even though the infinitesimal site is in some sense not nicely behaved in positive characteristics, have people continued to study it in this context? What kind of results have been obtained, and has it still been useful? I'm particularly interested in results about $D$-modules in positive characteristic (i.e. crystals in the infinitesimal site if $X/S$ is smooth), but I am also curious to see in which other directions progress has been made.
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up vote 15 down vote accepted

There is a paper of Ogus for 1975, "The cohomology of the infinitesimal site", in which he shows that if $X$ is proper over an algebraically closed field $k$ of char. $p$, and embeds into a smooth scheme over $k$, then the infinitesimal cohomology of $X$ coincides with etale cohomology with coefficients in $k$ (or more generally $W_n(k)$ if we work with the infinitesimal site of $X$ over $W_n(k)$).

Note that, since $k$ has char. $p$, we are talking about etale cohomology with mod $p^n$ coefficients, so this is "smaller" than the usual etale cohomology; it just picks up the "unit roots" of Frobenius. So the infinitesimal cohomology gives the unit root part of the crystalline cohomology.

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Matt - is there a nice picture for all the intermediate answers, ie cohomology of structure sheaf on site given by X modulo nth Frobenius neighborhood of the diagonal (or equivalently as a D-module with all the assorted partially divided power structures)? –  David Ben-Zvi Feb 19 '10 at 14:58
    
Dear BZ, Since pulling back via Frobenius takes one from $\mathcal D^{(m)}$-modules to $\mathcal D^{(m+1)}$-modules, and the structure sheaf pulls back to itself, does this mean that the answer is always the same for finite $m$ (i.e. always the crystalline cohomology)? –  Emerton Feb 19 '10 at 15:12
    
Thanks a lot, this looks very interesting. Maybe it would be too much to ask for, but do you know if there's also a log-version of this kind of analysis? –  Lars Feb 19 '10 at 22:46
    
Dear Lars, I'm not sure; maybe you could ask Ogus directly, since he is also a leading expert on log structures. –  Emerton Feb 20 '10 at 0:52
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