A prismatic approach to crystalline local systems

Let $X$ $X$ be a smooth $p$ $p$ -adic formal scheme. We show that integral crystalline local systems on the generic fiber of $X$ $X$ are equivalent to prismatic $F$ $F$ -crystals over the analytic locus of the prismatic site of $X$ $X$ . As an application, we give a prismatic proof of Fontaine’s ${\mathrm{C}}_{\mathrm{crys}}$ $\mathrm {C}_{{\mathrm {crys}}}$ -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 $F$ $F$ -crystals, including various comparison theorems, Poincaré duality, and Frobenius isogeny.