Crystalline condition for $A_{\inf }$-cohomology and ramification bounds

Abstract

For a prime $p$ and a smooth proper $p$-adic formal scheme $\mathcal{X}$ over ${\mathcal{O}}_K$ where $K$ is a $p$-adic field, we study a series of conditions (${\mathrm{Cr}}_s$), $s\ge 0$ that partially control the $G_K$‑action on the image of the associated Breuil–Kisin prismatic cohomology $\mathsf{R}{\Gamma }_{△△}(\mathcal{X}/\mathfrak{S})$ inside the $A_{\inf }$-prismatic cohomology $\mathsf{R}{\Gamma }_{△△}({\mathcal{X}}_{A_{\inf }}/A_{\inf }).$ The condition (${\mathrm{Cr}}_0$) is a crystallinity criterion for a Breuil–Kisin–Fargues $G_K$-module of Gee and Liu, and leads to a proof of crystallinity of ${\mathrm{H}}_{\text{eˊt}}^i({\mathcal{X}}_{\overline{\eta }},{\mathbb{Q}}_p)$ that avoids the crystalline comparison. Using the higher conditions (${\mathrm{Cr}}_s$), we are able to adapt the strategy of Caruso and Liu to establish ramification bounds for the mod $p$ representations ${\mathrm{H}}_{\text{eˊt}}^i({\mathcal{X}}_{\overline{\eta }},\mathbb{Z}/p\mathbb{Z})$, for arbitrarily large $e$ and $i$. This extends and/or improves existing bounds in various situations.