Suppose $\mathfrak{X}$ is a smooth projective scheme of finite type over $\mathbb{Z}_p$, with generic fibre $X$. Then there are comparison theorems relating de Rham and crystalline cohomology, $H^i_{\mathrm{dR}}(X / \mathbb{Q}_p) \cong H^i_{\mathrm{cris}}(\mathfrak{X}, \mathbb{Q}_p)$.

Does this work integrally, i.e. does this isomorphism match up $H^i_{\mathrm{dR}}(\mathfrak{X} / \mathbb{Z}_p)$ with $H^i_{\mathrm{cris}}(\mathfrak{X}, \mathbb{Z}_p)$?

Similarly: what if one introduces also the etale cohomology $H^i_{\mathrm{et}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Q}_p)$? This contains a natural lattice $H^i_{\mathrm{et}}(X_{\overline{\mathbb{Q}}_p}, \mathbb{Z}_p)$. Does this match up with $H^i_{\mathrm{cris}}(\mathfrak{X}, \mathbb{Z}_p)$, via Fontaine's functor $\mathbb{D}_{\rm cris}$?