I have a basic question concerning comparison of different cohomology theories. Let $X$ be a projective smooth (or just proper smooth) variety over a separably closed field $k$ of characteristic $p,$ which is not liftable to characteristic zero. 
Is there any relation between the de Rham cohomology $H^n(X,\Omega^{\bullet}\_X)$ and the $\ell$-adic cohomology? For example, do they have the same dimension (over $k$ and $Q\_l$ resp.)?



comparison of de Rham cohomology and etale cohomology
I believe the answer is no, that these two spaces need not have the same vector space dimension. Grothendieck here cites an example of Serre in a footnote on the last page; unfortunately, I don't have access to Serre's original paper at the moment.


[http://www.numdam.org/item?id=PMIHES\_1966\_*29*\_95\_0](http://www.numdam.org/item?id=PMIHES_1966__29__95_0)


