A prismatic approach to crystalline local systems

Let X be a smooth p -adic formal scheme. We show that integral crystalline local systems on the generic fiber of X are equivalent to prismatic F -crystals over the analytic locus of the prismatic site of X . As an application, we give a prismatic proof of Fontaine’s C crys -conjecture, for general coefficients, in the relative setting, and allowing ramified base fields. Along the way, we also establish various foundational results for the cohomology of prismatic F -crystals, including various comparison theorems, Poincaré duality, and Frobenius isogeny.