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

TL;DR

This paper develops a Dieudonné theory for $n$-smooth group schemes over $F_p$-algebras, establishing an equivalence with Dieudonné modules and proving smoothness properties of their moduli stacks, confirming conjectures of Drinfeld.

Contribution

It introduces a Dieudonné module framework for $n$-smooth group schemes and proves the smoothness of their moduli stacks, advancing the understanding of Frobenius analogues.

Findings

Category of $n$-smooth group schemes is equivalent to a subcategory of Dieudonné modules.

The moduli stack $ ext{Sm}_n$ is smooth over $F_p$.

The truncation morphism $ ext{Sm}_{n+1} o ext{Sm}_n$ is smooth and surjective.

Abstract

For all $n \geq 1$, there is a notion of $n$-smooth group scheme over any $\mathbb{F}_p$-algebra $R$, which may be thought of as a ``Frobenius analogue" of $n$-truncated Barsotti-Tate groups over $R$. We show 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.