A prismatic approach to crystalline local systems

TL;DR

This paper establishes an equivalence between crystalline local systems and prismatic F-crystals on p-adic formal schemes, providing new proofs and foundational results for p-adic Hodge theory.

Contribution

It introduces a prismatic framework for crystalline local systems and proves Fontaine's C_crys-conjecture in a general, ramified setting.

Findings

Equivalence between integral crystalline local systems and prismatic F-crystals.

A prismatic proof of Fontaine's C_crys-conjecture.

Foundational cohomology results including comparison theorems and Poincaré duality.

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 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, Poincar\'e duality, and Frobenius isogeny.