Prismatic crystals for smooth schemes in characteristic $p$ with Frobenius lifting mod $p^2$

Abstract

Let $(A,(p))$ be a crystalline prism with $A_n = A/p^{n+1}A$ for all $n\geq 0$. Let $\frakX_0$ be a smooth scheme over $A_0$. Suppose that $\frakX_0$ admits a lifting $\frakX_n$ over $A_n$ and the absolute Frobenius $\rF_{\frakX_0}:\frakX_0\to \frakX_0$ 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 $(\frakX_0/A)_{\Prism}$ and the category of $p$-connections over $\frakX_n$, which is compatible with cohomologies. This generalises a previous work of Ogus. We also give some remarks on trivializing the Hodge--Tate gerbe $\pi_{\frakX_0}^{\rm HT}:\frakX_0^{\rm HT}\to\frakX_0$ introduced by Bhatt--Lurie.