We define, for each quasi-syntomic ring R (in the sense of Bhatt-Morrow-Scholze), a category DF(R) of filtered prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to DF(R). We prove that this functor is an antiequivalence when moreover R is flat over Z/p n for some n > 0 or over Zp. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze. JOHANNES ANSCH ÜTZ AND ARTHUR-C ÉSAR LE BRAS 5.2. Comparison over O K 5.3. Filtered prismatic Dieudonné crystals and displays 5.4. Étale comparison for p-divisible groups Appendix A. Descent for p-completely faithfully flat morphisms References 1 This means that the complex M 1,0] (R/p) for any R/p-module N .

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We define, for each quasisyntomic ring R (in the sense of Bhatt et al., Publ. Math. IHES129 (2019), 199–310), a category $\mathrm {DM}^{\mathrm {adm}}(R)$ of admissible prismatic Dieudonné crystals over R and a functor from p-divisible groups over R to $\mathrm {DM}^{\mathrm {adm}}(R)$ . We prove that this functor is an antiequivalence. Our main cohomological tool is the prismatic formalism recently developed by Bhatt and Scholze.

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