Dieudonné theory for $n$-smooth group schemes

Abstract

For all $n\ge 1$, 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 Dieudonné 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.

19 pages, revised exposition and minor corrections. To appear in Algebra & Number Theory