Inspired by a theorem of Bhatt-Morrow-Scholze, we develop a stacky approach to crystals and isocrystals on Frobenius-smooth schemes over F_p . This class of schemes goes back to Berthelot-Messing and contains all smooth schemes over perfect fields of characteristic p. To treat isocrystals, we prove some descent theorems for sheaves of Banachian modules, which could be interesting in their own right.
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