# Prismatic Dieudonn\'e theory

## TL;DR

This paper introduces a new category of admissible prismatic Dieudonné crystals for quasi-syntomic rings and proves an equivalence with p-divisible groups, utilizing prismatic cohomology techniques.

## Contribution

It defines a novel category of prismatic Dieudonné crystals and establishes an anti-equivalence with p-divisible groups over quasi-syntomic rings.

## Key findings

- Established an anti-equivalence between p-divisible groups and prismatic Dieudonné crystals.
- Developed a new cohomological framework using prismatic formalism.
- Extended Dieudonné theory to the setting of quasi-syntomic rings.

## Abstract

We define, for each quasi-syntomic ring $R$ (in the sense of Bhatt-Morrow-Scholze), a category $\mathrm{DM}^{\rm adm}(R)$ of \textit{admissible prismatic Dieudonn\'e crystals over $R$} and a natural functor from $p$-divisible groups over $R$ to $\mathrm{DM}^{\rm adm}(R)$. We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.