Spahn display of a formal p-divisible group

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’’:

$$w_n(F(x))=w_{n+1}(x),\; n\ge 0$$

$$w_n(V(x))=w_{n-1}(x),\; n\gt 0$$

$$w_0(V(x))=0$$

$$FV=p$$

$$VF(xy)=xV(y)$$

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 $P /ker(w_0)P$.

2. $V^{-1} : Q \to P$ is a $F$-linear epimorphism.

3. For $x \in P$ and $w \in W (R)$, we have $V^{-1} ( V wx) = wF x$.

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

• T. Zink, Windows for displays of p-divisible groups. in:Moduli of Abelian Varieties, Progress in Mathematics 195, Birkhäuser 2001, pdf