Dieudonn\'e theory over semiperfect rings and perfectoid rings

TL;DR

This paper extends Dieudonné theory to semiperfect and perfectoid rings, establishing categorical equivalences and classifying p-divisible groups via Breuil-Kisin-Fargues modules for p>2.

Contribution

It introduces a new window structure on the Dieudonné crystal over semiperfect rings and classifies p-divisible groups over perfectoid rings using Breuil-Kisin-Fargues modules.

Findings

Categorical equivalence under boundedness conditions

Classification of p-divisible groups over perfectoid rings

Extension of Dieudonné theory to broader ring classes

Abstract

The Dieudonn\'e crystal of a p-divisible group over a semiperfect ring R can be endowed with a window structure. If R satisfies a boundedness condition, this construction gives an equivalence of categories. As an application one obtains a classification of p-divisible groups and commutative finite locally free p-group schemes over perfectoid rings by Breuil-Kisin-Fargues modules if p>2.