What is the exact formulation of the Poincaré duality in the context of crystalline cohomology? and what is the best reference for this subject (which contains the proof too)? In Berthelot-Ogus and in Illusie's notes there is no formulation of this duality, although they use this.

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    $\begingroup$ The standard reference is Berthelot's "Cohomologie cristalline des schemas de caracterestique p > 0". $\endgroup$ – naf Jun 22 '11 at 13:35
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    $\begingroup$ NB. Crystalline cohomology is only good for proper smooth varieties. If you want a good Poincaré duality (for smooth varieties, maybe non proper), rigid cohomology is the thing to look at. See Pierre Berthelot, Dualité de Poincaré et formule de Künneth en cohomologie rigide (1997), Compte-rendus de l'Académie des sciences, vol. 325, no. 5, 493--498. $\endgroup$ – ACL Jun 28 '11 at 23:07

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