Dieudonné theory for $n$-smooth group schemes

For all $n\ge 1$, 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 Dieudonné modules over $R$. As a consequence, we show that the moduli stack ${\mathrm{S}\mathrm{m}}_n$ of $n$-smooth commutative group schemes is smooth over ${\mathbb{F}}_p$ and that the natural truncation morphism ${\mathrm{S}\mathrm{m}}_{n+1}\to {\mathrm{S}\mathrm{m}}_n$ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld.