# Poincare duality for algebraic de Rham cohomology with integrable connection coefficients

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!

• 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. Aug 5 at 14:17
• @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. Aug 6 at 1:14