I am reading "The Gauss-Manin Connection and Tannaka Duality" (here is the link to the paper). I am specifically interested in the proof of Proposition 2.2. In this proof, the authors use the Poincaré duality for algebraic de Rham cohomology of a smooth projective curve $X$ over a field $K$ of characteristic $0$ with integrable connection coefficient $(\mathcal V,\bigtriangledown)$. You can see this line lies between equations (2.12) and (2.13) in the paper:

$H^2_{dR}(X,(\mathcal V, \bigtriangledown)) $ is a Poincaré dual to $H^0_{dR}(X,(\mathcal V,\bigtriangledown)^\vee)$, where $(\mathcal V, \bigtriangledown)^\vee$ is the dual connection.

However, I can not find any references for proof of this claim. In the Stacks project (here), this duality is only proven for trivial connections $(\Omega ^ p_{X/k},d)$. So I wonder whether any of you have a reference for this more general duality.

Thanks a lot!

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    $\begingroup$ I have seen Proposition $2.5$, page$7$ here, where Esnault summarises duality results ("reminders on duality"). I don't know if it applies. Perhaps there are other reminders in some of her papers. $\endgroup$ Aug 5 at 14:17
  • $\begingroup$ @Dietrich Burde The duality you mention is for etale cohomology. I have searched more in her other papers but it seems that she did not mention it anywhere. $\endgroup$
    – Khainq
    Aug 6 at 1:14

1 Answer 1


It appears in the book "De Rham Cohomology of Differential Modules on Algebraic Varieties" of Yves André, Francesco Baldassarri, and Maurizio Cailotto, Theorem D.2.17.


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