## Idea In general an assignation of an object of linear algebra to a $p$-divisible group is called a [[Dieudonné theory]]. There is a Dieudonné theory assigning to a formal $p$-divisible group $X$ over an excellent [[p-adic ring]] $R$ an object called *a display*. On the display one can read off the structural equations for the [[Cartier module]] of $X$. ## Definition Let $R\in CRing$ a commutative unitary ring. Let $W(R)$ denote the ring of [[Witt vectors]] of $R$. Let $$w_n:\begin{cases} W(R)\to R \\ (x_0,\dots,x_i,\dots)\mapsto x_0^{p^n}+p x_1^{p^{n-1}}+ \dots + p^n x_n \end{cases}$$ denote the morphism of rings assigning to Witt vector its correspnding [[Witt polynomial]]. Let $$w_n:\begin{cases} W(R)\to W(R) \\ (x_0,\dots,x_i,\dots)\mapsto (0,x_0,\dots,x_i,\dots) \end{cases}$$ denote the [[Verschiebung morphism]] which is a morphism of the underlying additive groups. Let $p$ be a [[prime number]] and let $$F:W(R)\to W(R)$$ denote the [[Frobenius morphism]]. Then Frobenius, Verschiebung, and the Witt-polynomial morphism satisfy the ''$p$-adic Witt-Frobenius identities'': $$w_n(F(x))=w_{n+1}(x),\; n\ge 0$$ $$w_n(V(x))=w_{n-1}(x),\; n\gt 0$$ $$w_0(V(x))=0$$ $$FV=p$$ $$VF(xy)=xV(y)$$ +-- {: .num_defn} ###### Definition A *$3n$-display over R* is defined to be a quadruple $(P, Q, F, V^{-1})$ where $P$ is a finitely generated projective $W(R)$-module, $Q \subset P$ is a submodule and $F$ and $V^{-1}$ are $F$-linear maps $F : P \to P$, $V^{-1}: Q \to P$. The following properties are satisfied: 1. $ker(w_0)P \subset Q \subset P$ and $P /Q$ is a direct summand of the $W (R)$−module $P /ker(w_0)P$. 1. $V^{-1} : Q \to P$ is a $F$-linear epimorphism. 1. For $x \in P$ and $w \in W (R)$, we have $V^{-1} ( V wx) = wF x$. =-- +-- {: .num_defn} ###### Definition Let $(\alpha_{ij})$ be a invertible matrix satisfying a) $F e_j=\sum_{i=1}^h\alpha_{ij} e_i, \; j=1,\dots ,d$ b) $V^{-1} e_j=\sum_{i=1}^h\alpha_{ij} e_i, \; j=1,\dots ,h$ Let $(\beta_{kl})$ denote the inverse matrix of $(\alpha_{ij})$. Let $B:=(w_0(\beta_{kl})modulo p)_{k,l=d+1,\dots,k}$ let $B^{(p)}$ deonte the matrix obtained by raising all entries to the $p$-th power. $(\alpha_{ij})$ is said to satisfy *the $V$-nilpotence condition* if there is a natural number $N\in \mathbb{N}$ such that $B^{p^{N-1}}\cdot\dots\cdot B^{(p)}\cdot B=0$. Then a $3n$-display satisfying the $V$-nilpotence condition locally on the [[spectrum]] $Spec R$ is called a *display*. =-- ## Properties ## Examples ## References * T. Zink, the display of a formal p-divisible group, to appear in Asterisque, [pdf](http://www.math.uni-bielefeld.de/~zink/display.pdf) * T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, [pdf](http://www.math.uni-bielefeld.de/~zink/Texel.pdf)