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].


14L05 Formal groups, \(p\)-divisible groups
14F30 \(p\)-adic cohomology, crystalline cohomology