Relative $(\varphi,\Gamma)$-modules and prismatic $F$-crystals

TL;DR

This paper establishes an equivalence between prismatic F-crystals with inverted I and étale Z_p-local systems on the generic fiber of p-adic formal schemes, linking cohomological properties in p-adic geometry.

Contribution

It proves an equivalence of categories connecting prismatic F-crystals and étale local systems, advancing the understanding of p-adic Hodge theory.

Findings

Equivalence of categories for prismatic F-crystals and étale local systems

Comparison of cohomology theories in p-adic geometry

Extension of prismatic theory to formal schemes

Abstract

In this paper, we prove that for any $p$-adic smooth separated formal scheme $\mathfrak X$, the category of prismatic $F$-crystals with $I$ inverted is equivalent to the category of \'etale $\mathbb Z_p$-local systems on the generic fiber of $\mathfrak X$. We also compare the cohomology of the corresponding coefficients.