Prismatic Dieudonné theory

Abstract

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 .

A Note on Dieudonné Theory over Perfectoid Rings

“…The functor of the theorem above can Zariski-locally on R be defined as M(G) := coker(M(H 1 ) → M(H 2 )) using the anti-equivalence of Theorem 1.5 (cf. proof of [6] Theorem 10.12 and proof of [1] Theorem 5.1.4).…”

“…We recall some general facts about perfectoid rings and briefly state the antiequivalences obtained in [6] ( §9 and §10), [1] ( § 4 and §5.1) and [7] (Theorem 17.5.2) which characterize Barsotti-Tate groups and p-groups over perfectoid rings in terms of semilinear algebra.…”

“…Lau extended these results to perfect rings of characteristic p in [5], reproving unpublished results of Gabber by a different method. Recently, in [1], Anschütz and Le Bras generalized the existing instances of Dieudonné theory to so called Prismatic Dieudonné Theory, including results on Dieudonné theory for perfectoid rings, which were partially proven by Lau before in [6]. In this note, we use the results of [1] to classify BT n -groups over perfectoid rings in terms of semi-linear algebra and investigate the consequences for lifting properties of BT n -groups to BT-groups in this setting.…”

“…In the case n = 1 one additionally demands that coker(ϕ) is finite projective as an S/ξ-module and that the induced sequence M/ξ ψ − → M σ /ξ ϕ − → M/ξ is exact. The category of BK n -modules over R, denoted by BK n (R), also admits a fibrewise characterization: A torsion Breuil-Kisin-Fargues module over R is a BK n -module if and only if it is killed by p n and all fibres are truncated Dieudonné modules of level n. Combining this fact with the fibrewise description of BT n -groups and the properties of the functor of [1], which classifies commutative finite locally free p-groups, we show the following result: Notations and assumptions:…”

Dieudonné Theory via Cohomology of Classifying Stacks

“…In [SW14], Scholze and Weinstein defined a mixed characteristic analogue of (covariant) Dieudonné modules for p-divisible groups over a perfectoid ring. More recently, in [ALB19], Anschütz and Le Bras defined a mixed characteristic analogue of contraviariant Dieudonné modules over more general base rings.…”