Th\'eorie de Dieudonn\'e cristalline et p\'eriodes p-adiques

In this paper we describe the Dieudonn\'e crystal of a finite locally free group scheme with a vector action of a finite field $\mathbb{F}$. These $\mathbb{F}$-vector schemes appear when we consider torsion points of $p$-divisible modules. A particular class of $\mathbb{F}$-vector schemes has been classified by Raynaud in [Ray74], which allows us to determine the structure of torsion points of $p$-divisible modules, under certain conditions on the Lie algebra.

This paper is the augmented notes of a course I gave jointly with Laurent Berger in Rennes in 2014. Its aim was to introduce the periods rings B crys and B dR and state several comparison theorems between{\'e}tale and crystalline or de Rham cohomologies for p-adic varieties.

The paper contains a proof of the Fontaine-Jannsen conjecture based on a crystalline version of the p-adic Poincar'e lemma (different proofs were found earlier by Faltings, Niziol and Tsuji).

We discuss the relation between crystalline Dieudonne theory and Dieudonne displays, with special emphasis on the case p=2. The theory of Dieudonne displays is extended to this case without restriction, which implies that the classification of finite flat group schemes by Breuil-Kisin modules holds for p=2 as well.

In this paper, we prove two results: first, we use crystalline cohomology of classifying stacks to directly reconstruct the classical Dieudonn\'e module of a finite, $p$-power rank, commutative group scheme $G$ over a perfect field $k$ of characteristic $p>0$. As a consequence, we give a new proof of the isomorphism $\sigma^* M(G) \simeq \mathrm{Ext}^1 (G, \mathcal{O}^{\mathrm{crys}})$ due to Berthelot--Breen--Messing using stacky methods combined with the theory of de Rham--...

Let $k$ be a perfect field of characteristic $p>2$ and $K$ an extension of $F=\mathrm{Frac} W(k)$ contained in some $F(\mu_{p^r})$. Using crystalline Dieudonn\'e theory, we provide a classification of $p$-divisible groups over $\mathscr{O}_K$ in terms of finite height $(\varphi,\Gamma)$-modules over $\mathfrak{S}:=W(k)[[u]]$. Although such a classification is a consequence of (a special case of) the theory of Kisin--Ren, our construction gives an independent proof and allows ...

This paper studies the derived de Rham cohomology of F_p and p-adic schemes, and is inspired by Beilinson's recent work. Generalising work of Illusie, we construct a natural isomorphism between derived de Rham cohomology and crystalline cohomology for lci maps of such schemes, as well logarithmic variants. These comparisons give derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory. Placing these ideas in the skeleton of Beilinson's co...

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 $H^2_{crys}(BG)$ recovers the Dieudonn\'e module of $G$. We also provide a calculation of crystalline cohomology of classifying stack of abelian varieties. We use this to prove that crystalline cohomology of the classifying stack of a $p$-divisible group is a symmetric algebra (in degree $2$) on its Di...

Let $\mathcal{O}_{K}$ be a complete discrete valuation ring of mixed characteristic with perfect residue field, endowed with its canonical log-structure. We prove that log $p$-divisible groups over $\mathcal{O}_{K}$ correspond to Dieudonn\'e crystals on the absolute log-prismatic site of $\mathcal{O}_{K}$ endowed with the Kummer log-flat topology. The proof uses log-descent to reduce the problem to the classical prismatic correspondence, recently established by Ansch\"utz-Le ...

We continue our study on the corresponding period rings with big coefficients, with the corresponding application in mind on relative $p$-adic Hodge theory and noncommutative analytic geometry. In this article, we extend the discussion of the corresponding noncommutative descent over \'etale topology to the corresponding noncommutative descent over pro-\'etale topology in both Tate and analytic setting.