Prismatic Dieudonn\'e theory

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.