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arXiv · 2406.08259

A Tannakian framework for prismatic $F$-crystals

Naoki Imai·Hiroki Kato·Alex Youcis
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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.

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BibTeXRIS

Naoki Imai, Hiroki Kato, Alex Youcis. 2024-06-12. A Tannakian framework for prismatic $F$-crystals. https://doi.org/10.1017/fms.2026.10228

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