Dieudonn\'e theory for $n$-smooth group schemes

Abstract

For all $n \geq 1$, there is a notion of an $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a ``Frobenius analogue" of an $n$-truncated Barsotti--Tate group over $R$. We prove that the category of $n$-smooth commutative group schemes over $R$ is equivalent to a certain full subcategory of Dieudonn\'e modules over $R$. As a consequence, we show that the moduli stack $\mathrm{Sm}_n$ of $n$-smooth commutative group schemes is smooth over $\mathbb{F}_p$ and that the natural truncation morphism $\mathrm{Sm}_{n+1} \to \mathrm{Sm}_n$ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld.