Dieudonn\'e theory for n-smooth group schemes

For all n ≥ 1, there is a notion of n-smooth group scheme over any Fₚ-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 Smₙ of n-smooth commutative group schemes is smooth over Fₚ and that the natural truncation morphism Smₙ₊₁ → Smₙ is smooth and surjective. These results affirmatively answer conjectures of Drinfeld.