In general an assignment of an object of linear algebra to a -divisible group is called a Dieudonné theory?.
There is a Dieudonné theory assigning to a formal -divisible group over an excellent p-adic ring? an object called a display. On the display one can read oﬀ the structural equations for the Cartier module of .
Let a commutative unitary ring. Let denote the ring of Witt vectors of . Let
denote the morphism of rings assigning to Witt vector its correspnding Witt polynomial. Let
denote the Verschiebung morphism? which is a morphism of the underlying additive groups. Let be a prime number and let
denote the Frobenius morphism. Then Frobenius, Verschiebung, and the Witt-polynomial morphism satisfy the ‘’-adic Witt-Frobenius identities’’:
A -display over R is defined to be a quadruple where is a ﬁnitely generated projective -module, is a submodule and and are -linear maps , .
The following properties are satisﬁed:
and is a direct summand of the −module .
is a -linear epimorphism.
For and , we have .
Let be a invertible matrix satisfying
Let denote the inverse matrix of . Let let deonte the matrix obtained by raising all entries to the -th power. is said to satisfy the -nilpotence condition if there is a natural number such that .
Then a -display satisfying the -nilpotence condition locally on the spectrum is called a display.
The following theorem compares the Dieudonné theory in terms of displays and the crystalline Dieudonné theory? of Grothendieck and Messing Messing.
Let be a display over . Then there is an isomorphism of crystals
where on the right side is the crystal defined in Messing.
Lawrence Breen, Pierre Berthelot, William Messing, Théorie de Dieudonné cristalline II
T. Zink, the display of a formal p-divisible group, to appear in Astérisque, pdf
T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, pdf