A Tannakian framework for prismatic F -crystals

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.