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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 $\mathbb{D}_{\rm cris}$?

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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) –  jnewton Nov 18 '10 at 13:12
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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 $e < p-1$, which is to say $p > 2$, 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 $t^p \in p A_{\rm{cris}}$. –  BCnrd Nov 18 '10 at 14:57
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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. –  BCnrd Nov 18 '10 at 16:01
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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). –  Kevin Buzzard Nov 18 '10 at 18:06
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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. –  ACL Nov 21 '10 at 0:13
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