It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas Weil cohomology theories should take values in characteristic $0$. It turns out that de Rham cohomology picks up torsion from Crystalline cohomology, so the dimensions are actually not the correct ones.


However, even if it's not a Weil cohomology theory, one could still try to prove some of the properties. The one I am interested in is Poincaré duality, including the construction of the cup product pairing, say for smooth proper varieties.


I think I know how to do this, but it is a bit technical. I was wondering if there is a place in the literature where it is carried out. It seems that most sources on algebraic de Rham cohomology immediately assume characteristic $0$, and don't prove the results for characteristic $p$ (presumably because de Rham cohomology is 'not the correct thing to consider').


**Edit.** Maybe I should have stated more clearly what I already know, and what it is that I am looking for. I am certainly convinced of the truth of the statement. For $X$ projective, there is a nice Čech-theoretic approach, worked out below by David Speyer. In general, there is a dévissage approach where we truncate the de Rham complex and induct on the number of terms. One place where one can read about this is [these notes (p. 3-4)](http://math.columbia.edu/~dejong/seminar/note_on_algebraic_de_Rham_cohomology.pdf) by Johan de Jong. In the notes, it is only carried out for $X$ projective, but the same approach should work for $X$ proper. (The notes assume characteristic $0$, but this is not used in the construction.)


However, the reason I posted this reference request is that I want to use the result in a paper. I prefer using peer reviewed sources to informal notes, and this is the type of result that should have been known for more than 30 years. I was hoping someone happens to know where to find it.



Reference for de Rham cohomology in positive characteristic
Prompted by user Yai0Phah's comment, let me answer my question: I did indeed find a reference. It turns out the correct place to look for these things is any book on crystalline cohomology, because
$$H^i\_{\operatorname{dR}}(X) = H^i\_{\operatorname{cris}}(X/k).$$
One canonical reference for Poincaré duality for smooth proper varieties in the more general setting of crystalline cohomology over $W\_n(k)$ is [Berthelot, Ch. VII, Thm. 2.1.3]. The proof for $H^\*\_{\operatorname{cris}}(X/W\_n(k))$ actually immediately reduces to the case of $H^\*\_{\operatorname{cris}}(X/k)$. This is then carried out by a careful dévissage on the successive truncations $\tau\_{\geq i} \Omega\_X^\*$ and $\tau\_{\leq n-i} \Omega\_X^\*$, which in the end deduces the statement from Serre duality. $\square$




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**References.**


[Berthelot] Berthelot, Pierre, *Cohomologie cristalline des schemas de caractéristique $p > 0$*, Lecture Notes in Mathematics **407** (1974). [ZBL0298.14012](https://zbmath.org/?q=an:0298.14012).


