Keerthi: I am not yet sure why Asterisque 223 shows that $\mathcal{O}^{cris}_n(\mathcal{O}_{\bar K}/p)=\mathcal{O}_{\bar K}/p^n$. Probably I am missing some important fact, but as I understand Asterisque 223 this object should be some $W_n(R_{\mathcal{O}_{\bar K}/p})$. Also, the universal PD-thickening only exists if the Frobenius on $A$ is surjective. But Fontains remark II.1.4 says that he constructs an morphism $W_n(A) \to \mathcal{O}^{cris}_n(A)$ for every $A$, it just may not give rise to an isomorphism. But how should this morphism be constructed as long as we don't know $O_n^{cris}(A)$?

Hi! Thank you both for your answers. I wasn't aware of the universal PD thickenings presented in Périodes $p$-adiques. That helps me a lot. Filippo: In II.1.4 (not in I.1.4) Fontaine does this contruction for all $k$-algebras, but his constructions gives an isomorphism only if the frobenius is surjective on A. But his should be possible with the constructions in Périodes $p$-adiques.