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.)?
[Math] comparison of de Rham cohomology and etale cohomology
ag.algebraic-geometryetale-cohomology
Best Answer
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