On Fontaine’s conjecture for torsion crystalline local systems

Abstract Let 𝔛 be a smooth connected 𝑝-adic formal scheme. Based on the prismatic description of crystalline local systems, we prove an analogue of Fontaine’s conjecture for torsion crystalline local systems on the generic fiber of 𝔛. As an application, we show that the locus of crystalline local systems whose Hodge–Tate weights lie in a fixed interval cuts out a closed subscheme of the universal deformation ring.