Dieudonné theory via cohomology of classifying stacks
Forum of Mathematics, Sigma, Tome 9 (2021)
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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.
@article{10_1017_fms_2021_77,
author = {Shubhodip Mondal},
title = {Dieudonn\'e theory via cohomology of classifying stacks},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {9},
year = {2021},
doi = {10.1017/fms.2021.77},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2021.77/}
}
Shubhodip Mondal. Dieudonné theory via cohomology of classifying stacks. Forum of Mathematics, Sigma, Tome 9 (2021). doi: 10.1017/fms.2021.77
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