Title: Can one compare integral structures on de Rham and crystalline cohomology?
Tags: nt.number-theory;ag.algebraic-geometry;etale-cohomology
Question: Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology,
$H^i_{\mathrm{dR}}(X / \mathbb{Q}_p) \cong H^i_{\mathrm{cris}}(\mathfrak{X}, \mathbb{Q}_p)$.

Does this work integrally, i.e. does this isomorphism match up $H^i_{\mathrm{dR}}(\mathfrak{X} / \mathbb{Z}_p)$ with $H^i_{\mathrm{cris}}(\mathfrak{X}, \mathbb{Z}_p)$? 

Similarly: what if one introduces also the etale cohomology $H^i_{\mathrm{et}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Q}_p)$? This contains a natural lattice $H^i_{\mathrm{et}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Z}_p)$. Does this match up with $H^i_{\mathrm{cris}}(\mathfrak{X}, \mathbb{Z}_p)$, via Fontaine's functor $\mathbb{D}_{\rm cris}$? 

Comment: And you don't really need Fontain's theory for curves.:)

Comment: Probably you mean to use ${\rm{H}}^i(\mathfrak{X}_0/\mathbf{Z}_p)$ (i.e., cohomology of special fiber, relative to $\mathbf{Z}_p$ as a PD-thickening of the residue field).  The integral comparison isom is valid if $e  2$, since then the divided powers are top. nilpotent.  The integral comparison morphism underlies the one with $p$ inverted; it is in the book of Berthelot and Ogus on crystalline cohom. I think that the etale-crystalline case (using $A_{\rm{cris}}$, right?) fails for $i = 2 \dim X \ge p$ since $t^p \in p A_{\rm{cris}}$.

Comment: Dear David: if you're just interested in curves then things become a lot more concrete (and simpler to prove, though $(1/2)\infty = \infty$...).  Presumably the degree-1 cohomology is what you care about, and so if $J$ denotes the relative Jacobian then the crystalline cohomology is the (contradvariant) Dieudonne module of the $p$-divisible group $G$ of $J_0$ whereas the etale cohomology is the $p$-adic Tate module of the generic fiber of $J$ (or of $G$).  So you're really asking about comparison for $p$-divisible groups in the absolutely unramified case, yes? If so, I can dig up a reference. 

Comment: Dear David: Do you care specifically about how the $p$-adic comparison morphism with $B_{\rm{cris}}$ behaves, or just if there is some "reasonable" setup which makes the integral structures match for curves?  The point is that the use of $A_{\rm{cris}}$ could be considered as a bit "arbitrary" insofar as there are plenty of other choices of period rings which give the same $\mathbf{Q}_p$-theory (e.g., in Fontaine's original 1982 Annals paper he didn't have $B_{\rm{cris}}$ but axiomatixed what he needed to get a reasonable $\mathbf{Q}_p$-theory and verified stuff for $p$-divisible groups).

Comment: @jnewton: As BCnrd says, one can define `$\mathbb{D}_{\mathrm{cris}}(T)$` for $T$ a finitely-generated `$\mathbb{Z}_p$`-module with an action of Galois, using Fontaine's ring `$\mathbb{A}_{\mathrm{cris}}$`.

Comment: @BCnrd: Thanks, that is really useful. I am mainly interested in curves, so $2 \mathrm{dim}(X) < p$ is a pretty mild condition :-)

Comment: Won't applying Fontaine's $\mathbb{D}_{\rm cris}$ to $H^i_{\mathrm{et}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Z}_p)$ forget the integral structure? (since $B_{\rm cris}$ is a $\mathbb{Q}_p$-algebra)

Comment: Hey David. To get some sort of feeling as to what is going on (and as people have pointed out it's not as easy as one might hope in small characteristic) you could do worse than look at the introduction to "F-isocrystals and De Rham Cohomology I" by Berthelot and Ogus (Inventiones 72, 1983).

Comment: Faltings (Integral crystalline cohomology over very ramified valuation rings, J. American Math. Soc. (1999) 12:1, p. 117-144,) 
has proved integral versions of the comparison theorems. I am more familiar with the case of $H^1$ where the first result in that direction is Fontaine's ``presque décomposition de Hodge-Tate''
(Formes différentielles et modules de Tate des variétés abéliennes sur les corps locaux, Invent. Math., t. 65, 1982, p. 379-409), 
see also my Bulletin of SMF paper (1998) for crystalline aspects
using Berthelot-Breen-Messing's crystalline Dieudonné theory.

Comment: Concerning the first part of your question: Kedlaya's algorithm computing the rigid cohomology, as well as its successors, all need to take care of integral structures. However, the cases treated in that litterature are quite specific.



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