## The display of a formal $$p$$-divisible group.(English)Zbl 1008.14008

Berthelot, Pierre (ed.) et al., Cohomologies $$p$$-adiques et applications arithmétiques (I). Paris: Société Mathématique de France. Astérisque. 278, 127-248 (2002).
Let $$R$$ be an excellent $$p$$-adic ring, $$W(R)$$ the ring of Witt vectors with coefficients in $$R$$. The author constructs a correspondence from objects called displays to formal $$p$$-divisible groups over $$R$$. A display is a quadruple $$(P,Q,F,V^{-1})$$ where $$P$$ is a finitely generated projective $$W(R)$$-module, $$Q\subset P$$ is a submodule, and $$F:P\rightarrow P$$ and $$V^{-1}:Q\rightarrow P$$ are $$^{F}$$-linear homomorphisms (where $$^{F}$$ is the Frobenius map) satisfying a condition called the $$V$$-nilpotence condition. If $$p$$ is nilpotent, then this correspondence gives a categorical equivalence between displays and formal $$p$$-divisible groups.
In the case $$R$$ is a perfect field then a formal group with Dieudonné module $$M$$ gives rise to the display $$(M,VM,F,V^{-1})$$ if $$V:M/pM\rightarrow M/pM$$ is nilpotent.
Given a display $$\mathcal{P}$$, the corresponding formal group $$BT_{\mathcal{P }}$$ is realized as the cokernel of the map $$V^{-1}-$$id$$:\mathbf{G}_{\mathcal{ P}}^{-1}\rightarrow \mathbf{G}_{\mathcal{P}}^{0},$$ where $$\mathbf{G}_{ \mathcal{P}}^{0}(\mathcal{N})=\widehat{W}(\mathcal{N})\otimes _{W(R)}P$$ for $$\mathcal{N}$$ a nilpotent $$R$$-algebra and $$\mathbf{G}_{\mathcal{P}}^{-1}$$ is a certain subgroup of $$\mathbf{G}_{\mathcal{P}}^{0}.$$ Under this correspondence the height of $$BT_{\mathcal{P}}$$ is the same as the rank of $$P$$.
Finally, equivalence is shown between the new theory constructed here and crystalline Dieudonné theory.
The author claims the following: In the case where $$R$$ is an artinian local ring with perfect residue field of characteristic $$p>2$$, then the category of displays is equivalent to $$p$$-divisible groups over $$R$$. This conclusion still holds if the residue field has characteristic $$2$$ and $$2R=0.$$ The proof of this does not appear here, rather it will be given in a future paper.
For the entire collection see [Zbl 0990.00019].

### MSC:

 14L05 Formal groups, $$p$$-divisible groups 14F30 $$p$$-adic cohomology, crystalline cohomology