The Dieudonné module ( M(X) ) of a ( p )-divisible group ( X ) over a perfect field ( k ) of characteristic ( p ) can indeed be interpreted as a crystalline cohomology group in specific contexts, particularly when ( X ) arises from a scheme. Here's a structured explanation:

1. Mazur-Messing-Oda Result and Abelian Varieties

For an abelian variety ( A ) over ( k ), the ( p )-d...

Yes. There is a completely rigorous (and standard) way to see the Dieudonné module of a p-divisible group as a cohomology group: it is an Ext^1 in the crystalline topos of the base.

Let S be a scheme on which p is locally nilpotent (e.g. S = Spec k for a perfect field k of characteristic p), and let G be a p-divisible group over S. Berthelot–Breen–Messing construct the (covariant) Dieudonné cryst...

This is a very interesting question about the categorical/cohomological nature of the Dieudonné module functor. Let me address both the philosophical and technical aspects.

The Cohomological Interpretation

Yes, there are rigorous ways to interpret the Dieudonné module as a cohomology theory, though the situation is subtle.

1. The Crystalline Realization

The most direct approach is throu...

Yes – after one passes from the module to the (contravariant) Dieudonné crystal, the relation with “cohomology’’ is completely literal.
I will try to explain the precise statement, the construction, and how it specialises to the equalities of Mazur–Messing–Oda.

Notation.
S is a scheme on which p is locally nilpotent; if you prefer, take S = Spec k with k perfect of characteristic p. ...

One way to see “Dieudonné‐modules are really cohomology” is to observe that Messing’s Dieudonné‐crystal is by definition an Ext–group in the crystalline topos of the base. I will sketch two (equivalent) points of view:

1. The Ext–definition (Messing, BBM).
   Let S be a p‐adic base (say S=Spec k with k perfect of char p, or more generally any p‐nilpotent formal scheme) and let X→S be a p‐divis...

<think> Okay, the user is asking whether the Dieudonné module can be rigorously interpreted as a cohomology group. They mention the Mazur-Messing-Oda result linking it to crystalline cohomology for abelian varieties but want a broader categorical framework.

Hmm, this is a deep question in arithmetic geometry. The user seems quite knowledgeable - they're referencing advanced concepts like F-cryst...

Yes. What is usually called “the Dieudonné module’’ of a (p)-divisible group is literally the value of a certain 1-st cohomology/Ext group on the crystalline site, and the classical formula
(M(A[p^{\infty}]);=;H ^1_{\rm crys}(A/W(k))) is only the special case in which the (p)-divisible group happens to be the (p)-power torsion of an abelian scheme.

Below is a concise description of t...

This is an excellent question that gets to the heart of how modern arithmetic geometry views -adic cohomology theories. The short answer is yes, the Dieudonné module can be rigorously interpreted as a cohomology group, and this interpretation is a cornerstone of the theory. The key is to use the correct site, which turns out to be the log-crystalline site.

Let's break down how this wor...

Is the Dieudonné module actually a cohomology group?

This is an excellent question about the cohomological interpretation of Dieudonné modules.

Rigorous cohomological interpretations

There are indeed several ways to interpret the Dieudonné module cohomologically:

1. Via crystalline site: For a -divisible group over a perfect field , one can define a crystalline site $\text{C...