Dieudonn\'e theory via cohomology classifying stacks II

In this paper, we prove two results: first, we use crystalline cohomology of classifying stacks to directly reconstruct the classical Dieudonn\'e module of a finite, p-power rank, commutative group scheme G over a perfect field k of characteristic $p>0$. As a consequence, we give a new proof of the isomorphism σ^* M(G) Ext¹ (G, Oᶜʳʸˢ) due to Berthelot--Breen--Messing using stacky methods combined with the theory of de Rham--Witt complexes. Additionally, we show that finite locally free commutative group schemes of p-power rank over a quasisyntomic base ring embed fully faithfully into the category of prismatic F-gauges, which extends the work of Ansch\"utz and Le Bras on prismatic Dieudonn\'e theory.