Let (A, (p)) be a crystalline prism with \(A_n = A/p^{n+1}A\) for all \(n\ge 0\) . Let \({\mathfrak {X}}_0\) be a smooth scheme over \(A_0\) . Suppose that \({\mathfrak {X}}_0\) admits a smooth lifting \({\mathfrak {X}}_n\) over \(A_n\) and the absolute Frobenius \({\mathrm F}_{{\mathfrak {X}}_0}:{\mathfrak {X}}_0\rightarrow {\mathfrak {X}}_0\) admits a lifting over \(A_1\) . Then we show that there is an equivalence between the category of the prismatic crystals of truncation n on and the category of p-connections over \({\mathfrak {X}}_n\) , which is compatible with cohomologies. This generalises a previous work of Ogus. We also give some remarks on trivializing the Hodge–Tate gerbe \(\pi _{{\mathfrak {X}}_0}^\textrm{HT}:{\mathfrak {X}}_0^\textrm{HT}\rightarrow {\mathfrak {X}}_0\) introduced by Bhatt–Lurie.
