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Algebraic and Complex Geometry
Dieudonné theory via cohomology of classifying stacks
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- 15 December 2021, e81
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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
$H^2_{\text {crys}}(BG)$ recovers the Dieudonné module of G. We also provide a calculation of the crystalline cohomology of the classifying stack of an abelian variety. We use this to prove that the 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.
