A prismatic approach to crystalline local systems

Abstract

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 \(\mathrm {C}_{{\mathrm {crys}}}\)-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 \(F\)-crystals, including various comparison theorems, Poincaré duality, and Frobenius isogeny.