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Thank you for the answer to the first question. As to the second, for some reason I was thinking of crystalline cohomology. Would $k=\mathbb{Z}_p[[t]]$ work? I don't know how to answer regarding the prism structure. I mean cohomology of the structure sheaf $\mathcal{O}_{\Delta}$. Is there extra information that I have to give? Do I have comparison with crystalline and etale for all the choices of this extra information? I did not notice that while reading the notes.
