# Finite Field Varieties and the de Rham Complex of Kähler Differentials

In an answer to a previous question I asked, where the Kahler differentials of a variety over a finite field were discussed, it was stated that:

You can certainly define de Rham cohomology using Kähler differentials, but over a field of characteristic p>0 . . . it is somewhat pathological: the Poincaré lemma can fail etc

(1) What does this "etc" mean here? That is to say, what else goes wrong?

(2) Is the de Rham complex for Kähler differentials useful for anything? Or is it just a curiosity?

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The Hodge-de-Rham spectral sequence may not degenerate at E_2, and the Hodge symmetry $h^{pq}=h^{qp}$ may break down, etc. –  shenghao Jul 21 '12 at 23:20
I find it strange you put a bounty on this question. "Is the de Rham complex useful for anything?" If you don't know the answer to that, then it probably isn't useful to you. Do you have some reason you have taken a great interest in this? As pointed out, you always have the Hodge-de Rham spectral sequence which I've found incredibly useful (even if it doesn't degenerate you still get valuable information from it). The difference of the de Rham and $\ell$-adic Betti numbers measures torsion in the crystalline cohomology, etc ... –  Matt Aug 9 '12 at 0:17