A Tannakian framework for prismatic $F$-crystals

TL;DR

This paper establishes a Tannakian framework for prismatic $F$-crystals on smooth formal schemes, linking them to $ ext{Z}_p$-local systems and developing a shtuka realization functor with compatibility properties.

Contribution

It introduces a Tannakian equivalence for prismatic $F$-crystals and constructs a shtuka realization functor compatible with existing theories.

Findings

Equivalence between prismatic $F$-crystals and $ ext{Z}_p$-local systems.

Development of a shtuka realization functor.

Compatibility with previous Tannakian theories of shtukas.

Abstract

We develop the Tannakian theory of (analytic) prismatic $F$-crystals on a smooth formal scheme $\mathfrak{X}$ over the ring of integers of a discretely valued field with perfect residue field. Our main result gives an equivalence between the $\mathcal{G}$-objects of prismatic $F$-crystals on $\mathfrak{X}$ and $\mathcal{G}$-objects on a newly-defined category of $\mathbb{Z}_p$-local systems on $\mathfrak{X}_\eta$: those of prismatically good reduction. Additionally, we develop a shtuka realization functor for (analytic) prismatic $F$-crystals on $p$-adic (formal) schemes and show it satisfies several compatibilities with previous work on the Tannakian theory of shtukas over such objects.