Completed prismatic $F$-crystals and crystalline $\mathbf{Z}_p$-local systems

TL;DR

This paper establishes an equivalence between completed prismatic $F$-crystals and crystalline $ extbf{Z}_p$-local systems on the generic fiber of a smooth $p$-adic formal scheme, extending prior work to more general settings.

Contribution

It introduces completed $F$-crystals on the absolute prismatic site and proves their categorical equivalence with crystalline $ extbf{Z}_p$-local systems, generalizing Bhatt and Scholze's results.

Findings

Defined a functor from completed prismatic $F$-crystals to crystalline $ extbf{Z}_p$-local systems

Proved the functor is an equivalence of categories

Extended the theory to more general formal schemes

Abstract

We introduce the notion of completed $F$-crystals on the absolute prismatic site of a smooth $p$-adic formal scheme. We define a functor from the category of completed prismatic $F$-crystals to that of crystalline \'etale $\mathbf{Z}_p$-local systems on the generic fiber of the formal scheme and show that it gives an equivalence of categories. This generalizes the work of Bhatt and Scholze, which treats the case of a complete discrete valuation ring with perfect residue field.