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

For a prime p and a smooth proper p -adic formal scheme \mathscr{X} over \mathcal{O}_K where K is a p -adic field, we study a series of conditions ( \mathrm{Cr}_s ), s\geq 0 that partially control the G_K ‑action on the image of the associated Breuil–Kisin prismatic cohomology \def\prisma{\vartriangle\hspace*{-3.9pt}{\tiny\vartriangle}}\mathsf{R}\Gamma_{\prisma}(\mathscr{X}/\mathfrak{S})\ inside the A_{\mathrm{inf}} -prismatic cohomology \def\prisma{\vartriangle\hspace*{-3.9pt}{\tiny\vartriangle}}\mathsf{R}{\Gamma_{\prisma}}(\mathscr{X}_{A_{\mathrm{inf}}}/A_{\mathrm{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}^{i}_{\text{\'{e}t}}(\mathscr{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}^{i}_{\text{\'{e}t}}(\mathscr{X}_{\overline{\eta}},\mathbb{Z}/p\mathbb{Z}) , for arbitrarily large e and i . This extends and/or improves existing bounds in various situations.