If you want to compute crystalline cohomology of a smooth proper variety $X$ over a perfect field $k$ of characteristic $p$, the first thing you might want to try is to lift $X$ to the Witt ring $W\_k$ of $k$. If that succeeds, compute de Rham cohomology of the lift over $W\_k$ instead, which in general will be much easier to do. Neglecting torsion, this de Rham cohomology is the same as the crystalline cohomology of $X$.


I would like to have an example at hand where this approach fails: Can you give an example for



> 
> A smooth proper variety $X$ over the finite field with $p$ elements, such that there is no smooth proper scheme of finite type over $\mathbb Z\_p$ whose special fibre is $X$.
> 
> 
> 


The reason why such examples *have* to exist is metamathematical: If there werent any, the pain one undergoes constructing crystalline cohomology would be unnecessary.



Lifting varieties to characteristic zero.
This paper of Serre gives an example (I've justed pasted I. Barsotti's math-sci review).
(The paper can be found in Serre's "Collected Works vol. II 1960-1971)



> 
> Serre, Jean-Pierre Exemples de
>  variétés projectives en
>  caractéristique $p$ non relevables en
>  caractéristique zéro. (French) Proc.
>  Nat. Acad. Sci. U.S.A. 47 1961
>  108--109.
> 
> 
> An example of a non-singular
>  projective variety $X\_0$, over an
>  algebraically closed field $k$ of
>  characteristic $p$, which is not the
>  image, $\text{mod}\,p$, of any variety
>  $X$ over a complete local ring of
>  characteristic 0 with $k$ as residue
>  field. The variety $X\_0$ is obtained
>  by selecting, in a 5-dimensional
>  projective space $S$, and for $p>5$, a
>  non-singular variety $Y\_0$ which has
>  no fixed point for an abelian finite
>  subgroup $G$ with at least 5
>  generators of period $p$, of the group
>  $\Pi(k)$ of projective transformations
>  of $S$, but which is transformed into
>  itself by $G$; then $X\_0=Y\_0/G$. The
>  reason for the impossibility is that
>  $\Pi(K)$, for a $K$ of characteristic
>  0, does not contain a subgroup
>  isomorphic to $G$. {Misprint: on the
>  last line on p. 108 one should read
>  $s(\sigma)=\exp(h(\sigma)N)$.}
> 
> 
> 


