Dieudonné Theory via Cohomology of Classifying Stacks

Abstract

We prove that if $G$ is a finite flat group scheme of $p$ power rank over a perfect field of characteristic $p$, then the second crystalline cohomology of its classifying stack $H_{crys}^2(BG)$ recovers the Dieudonné module of $G$. We also provide a calculation of crystalline cohomology of classifying stack of abelian varieties. We use this to prove that crystalline cohomology of the classifying stack of a $p$-divisible group is a symmetric algebra (in degree $2$) on its Dieudonné module. We also prove mixed characteristic analogues of some of these results using prismatic cohomology.

arXiv comment21 Pages, added Theorem 1.4

References in corpus (4)

• Higher and derived stacks: a global overview
• Topological Hochschild homology and integral $p$-adic Hodge theory
• Hodge theory of classifying stacks
• Counterexamples to Hochschild--Kostant--Rosenberg in characteristic $p$