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

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.