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? 



Is there an integral version of Faltings' isomorphism in p-adic Hodge theory between etale and Hodge cohomologies
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.


