TITLE: Reference for de Rham cohomology in positive characteristic
QUESTION [25 upvotes]: 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) 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.

REPLY [4 votes]: 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. $\square$

References.
[Berthelot] Berthelot, Pierre, Cohomologie cristalline des schemas de caractéristique $p > 0$, Lecture Notes in Mathematics 407 (1974). ZBL0298.14012.