Dieudonné theory via classifying stacks and prismatic $F$-gauges
Shubhodip Mondal
TL;DR
This work unifies crystalline and prismatic perspectives on Dieudonné theory through a stack-theoretic lens. It shows that the classical Dieudonné module $M(G)$ can be recovered from $H^2_{ ext{crys}}(BG)$, and provides a new, concise proof of the Berthelot–Breen–Messing isomorphism using classifying stacks and de Rham–Witt theory. It then extends to mixed characteristic via prismatic cohomology by introducing prismatic Dieudonné $F$-gauges $\mathcal{M}(G)$, proving a fully faithful correspondence for $p$-divisible groups and finite locally free p-power rank group schemes over quasisyntomic bases, and establishing duality and connections to Galois representations. The approach also recovers and generalizes previous classification results for $p$-divisible groups, and provides a cohesive framework to compute cohomology with coefficients and to read off arithmetic information from prismatic Dieudonné gauges, including abelian schemes and their Tate modules. Overall, the paper builds a geometric, stack-theoretic realization of Dieudonné theory across equal and mixed characteristics, with broad implications for cohomology theories and Galois-representation correspondences.
Abstract
In this paper, we apply stack theoretic ideas to the classification problem in Dieudonné theory. First, we use crystalline cohomology of classifying stacks to directly reconstruct the classical Dieudonné 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, much shorter proof of the isomorphism $σ^* M(G) \simeq \mathrm{Ext}^1 (G, \mathcal{O}^{\mathrm{crys}})$ 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 can be classified in terms of ``prismatic Dieudonné $F$-gauges", which we introduce by making constructions using (higher) classifying stacks. The latter generalizes the result of Anschütz and Le Bras on classification of $p$-divisible groups, which we also reprove using our approach. Along the way, we prove a description of cohomology with coefficients in group schemes, compatibility with Cartier duality, and reconstruction of Galois representations in terms of our prismatic Dieudonné $F$-gauges.