Dieudonné theory for \(n\)-smooth group schemes

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 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.