A Note on Dieudonn\'{e} Theory over Perfectoid Rings

Abstract

For a perfectoid ring $R$ and a natural number $n$ we investigate the essential image of the category of truncated by $n$ Barsotti-Tate groups under the anti-equivalence between commutative, finite, locally free, $R$-group schemes of $p$-power order and torsion Breuil-Kisin-Fargues modules over $R$. We describe the associated semi-liner algebra data and show as a consequence that every $\text{BT}_n$-group over $R$ is the $p^n$-torsion of some $\text{BT}$-group.