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Let $K$ be a $p$-adic field, that is a complete discrete valuation ring of characteristic $0$ with a perfect residue field $k$ of characteristic $p > 0$ (to simplify one could also take $K$ to be a finite extension of $\mathbb{Q}_p$). Let $\mathbb{C}_K$ be the completion of a (fixed) algebraic closure $\overline{K}$ of $K$. Then one of Faltings' theorems in $p$-adic Hodge theory says for any smooth proper $K$-scheme $X$ there is a natural isomorphism of $\mathbb{C}_K$-semilinear $\mathrm{Gal}(\overline{K}/K) = \mathrm{Gal}(\mathbb{C}_K/K)$-representations

$\mathbb{C}_K \otimes_{\mathbb{Q}_p} H^n_{et}(X_{\overline{K}}, \mathbb{Q}_p) \cong \bigoplus_{q\in\mathbb{Z}} (\mathbb{C}_K(-q) \otimes_K H^{n - q}(X, \Omega_{X/K}^q)).$

Here $X_{\overline{K}}$ is the base change of $X$ to the algebraic closure, whereas $\mathbb{C}_K(s)$ stands for the usual $s$-th order Tate twist by the cyclotomic character describing the action of the absolute Galois group on the $p$-power roots of unity.

My question is: is there an integral version of the above isomorphism? Let me be more precise and explain what I mean by this: let $\mathcal{X}$ be a smooth proper scheme over the valuation ring $\mathcal{O}_K$ of $K$ and let $\mathcal{O}_{\overline{K}}$ be the valuation ring of the algebraic closure. Is there an isomorphism similar to the one above relating, say, $H^n_{et}(\mathcal{X}\times_{\mathcal{O}_K} \mathcal{O}_{\overline{K}}, \mathbb{Z}_p)$ and the $H^{n - q}(\mathcal{X}, \Omega^q_{\mathcal{X}/\mathcal{O}_K})$'s?

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Dear Kestutis, See e.g. the articles of Fontaine--Messing, Faltings, and Caruso mentioned in the comments to the answers below. Regards, –  Emerton May 30 '12 at 2:30

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up vote 9 down vote accepted

Dear Kestutis,

your question is integral in two ways: first of all, you would like to consider schemes over a whole DVR instead of the generic fiber only. Secondly, you would like to have a comparison theorem between two $\mathcal{O}_\overline{K}$-modules and not only between $K$-vector spaces. The two are tightly connected since $H_\text{dR}(\mathcal{X})\otimes K=H_\text{dR}(X)$, if $X$ denotes the generic fiber of $\mathcal{X}$.

As for the first question we have the so-called Crystalline Conjecture proven by Faltings and many others. It says that if $\mathcal{X}$ is proper and smooth over $\mathcal{O}_K$ (I keep your notations) with generic fiber $X$ and special fiber $\overline{\mathcal{X}}$, then $$ B_\text{cris}\otimes_{K_0}H_\text{cris}^m(\overline{\mathcal{X}})\cong B_\text{cris}\otimes_{\mathbb{Q}_p}H_\text{et}(X_\overline{K},\mathbb{Q}_p) $$ where $B_\text{cris}$ is Fontaine's ring of periods. If you extend scalars to $B_\text{dR}\supset B_\text{cris}$ you get back the comparison isomorphism that you quote because $B_\text{dR}$ admits a filtration whose associated graded ring is $\bigoplus_{q\in\mathbb{Z}}\mathbb{C}_K(q)$; and because of a basic result by Berthelot and Berthelot-Ogus saying that crystalline cohomology (which is the group $H_\text{cris}(\overline{\mathcal{X}})$ in my equation above and is a cohomology theory of the special fiber with coefficients in $W(k)$, the Witt vectors of the residue field $k$ of $\mathcal{O}_K$) is isomorphic to de Rham cohomology after you tensor both with $\text{Frac}(W(k))$. You could give a look at
1) J.-M. Fontaine, Représentations $p$-adiques semi-stables, in Périodes $p$-adiques, Astérisque 223, Section 6.1 for a discussion about the Crystalline Conjecture
2) P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton University Press, Chapter 7 (the very last theorem) for the isomorphism $H_\text{cris}\cong H_\text{dR}$
Let me also remark that much of the above can be generalized assuming only that $\mathcal{X}$ be semistable instead of smooth (by Tsuji).

The second question is more subtle. As I mentioned, already the isomorphism $H_\text{cris}(\overline{\mathcal{X}})=H_\text{dR}(\mathcal{X})$ is false in general: it is true if $K/\mathbb{Q}_p$ (or $\mathcal{O}_K/\mathbb{Z}_p$, equivalently) has absolute ramification index $e\leq p-1$. In this case, at least, we dispose of an interpretation of the de Rham cohomology of $\mathcal{X}$ in terms of crystalline cohomology. We also have an "integral structure" $A_\text{cris}\subseteq B_\text{cris}$ and one can hope to have something like $$ A_\text{cris}\otimes_{W(k)}H_\text{cris}^m(\overline{\mathcal{X}})\stackrel{?}{\cong} A_\text{cris}\otimes_{\mathbb{Z}_p}Hˆm_\text{et}(\mathcal{X}_{\mathcal{O}_\overline{K}},\mathbb{Z}_p) $$ The problem is that $A_\text{cris}$ (and even $A_\text{cris}[1/p]$!) is much smaller than $B_\text{cris}$: roughly speaking, it cannot detect "negative Hodge-Tate weights".

[EDIT] As Keerthi Madapusi Pera and Matthew Emerton commented, in some special cases the above isomorphism holds true. They give precise references in the comments.

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Such an isomorphism does exist if $e=1$ and $m<p$. See Fontaine-Messing, 'p-adic periods and p-adic etale cohomology'. In particular, if $X$ is an abelian variety over $W(k)$ and $p>2$, then it holds in all degrees. This is somewhat sharp. The result already fails for $p=2$ and $e=1$, and $X$ is an ordinary abelian variety. –  Keerthi Madapusi Pera May 30 '12 at 2:19
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Dear Filippo, Faltings's Johns Hopkins article also gives an integral comparison theorem when $e = 1$ in degrees lying in the Fontaine--Laffaile range. More recently, Xavier Caruso has proved an integral semistable comparison theorem when $e$ times the cohomological degree $< p-1$. In general this question is part of the subject of "integral $p$-adic Hodge theory", which is a very active area of research. Regards, Matthew –  Emerton May 30 '12 at 2:30
    
Dear Keerthi, Dear Matthew – thanks! As I said I am really not an expert, I apologize. –  Filippo Alberto Edoardo May 30 '12 at 2:35
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Dear Filippo, I think I can speak for Keerthi as well as myself in saying that there's no need to apologize! Best wishes, Matthew –  Emerton May 30 '12 at 3:57
    
Same here!...... –  Keerthi Madapusi Pera May 30 '12 at 4:06

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