I'm interested in $p$-adic cohomology theories now. I have learned that since de Rham cohomology behaves badly in char $p$, people invented crystalline cohomology in smooth cases and later rigid cohomology for the non-smooth case. They have many desired properties like finite dimensionality, cohomology dimension, Poincare duality, Kunneth formula(I'm not that sure for this, but some of them are proved in the articles of Illusie, Berthelot, Crew and Kedlaya.). And in smooth case, rigid cohomology coincides with crystalline(I'm also not sure for this).
My question is if both the theories are Weil cohomology theory, i.e., have all the desired properties? If it is the case, if we can replace crystalline with rigid thoroughly?
I have learned the basic ideas of both cohomology theories in survey articles, but to study the explicit technics and constructions in crystalline theory is not easy. At the same time I'm reading on rigid theory via the book of Stum and articles of Kedlaya(Fourier transforms and p-adic Weil II and Finiteness of rigid cohomology with coefficients).
