# nLab display of a 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 $R\in \mathrm{CRing}$ a commutative unitary ring. Let $W\left(R\right)$ denote the ring of Witt vectors of $R$. Let

${w}_{n}:\left\{\begin{array}{l}W\left(R\right)\to R\\ \left({x}_{0},\dots ,{x}_{i},\dots \right)↦{x}_{0}^{{p}^{n}}+p{x}_{1}^{{p}^{n-1}}+\dots +{p}^{n}{x}_{n}\end{array}$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

${w}_{n}:\left\{\begin{array}{l}W\left(R\right)\to W\left(R\right)\\ \left({x}_{0},\dots ,{x}_{i},\dots \right)↦\left(0,{x}_{0},\dots ,{x}_{i},\dots \right)\end{array}$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

$F:W\left(R\right)\to W\left(R\right)$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”:

${w}_{n}\left(F\left(x\right)\right)={w}_{n+1}\left(x\right),\phantom{\rule{thickmathspace}{0ex}}n\ge 0$w_n(F(x))=w_{n+1}(x),\; n\ge 0
${w}_{n}\left(V\left(x\right)\right)={w}_{n-1}\left(x\right),\phantom{\rule{thickmathspace}{0ex}}n>0$w_n(V(x))=w_{n-1}(x),\; n\gt 0
${w}_{0}\left(V\left(x\right)\right)=0$w_0(V(x))=0
$\mathrm{FV}=p$FV=p
$\mathrm{VF}\left(\mathrm{xy}\right)=\mathrm{xV}\left(y\right)$VF(xy)=xV(y)
###### Definition

A $3n$-display over R is defined to be a quadruple $\left(P,Q,F,{V}^{-1}\right)$ where $P$ is a ﬁnitely generated projective $W\left(R\right)$-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. $\mathrm{ker}\left({w}_{0}\right)P\subset Q\subset P$ and $P/Q$ is a direct summand of the $W\left(R\right)$−module $P/\mathrm{ker}\left({w}_{0}\right)P$.

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

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

###### Definition

Let $\left({\alpha }_{\mathrm{ij}}\right)$ be a invertible matrix satisfying

a) $F{e}_{j}={\sum }_{i=1}^{h}{\alpha }_{\mathrm{ij}}{e}_{i},\phantom{\rule{thickmathspace}{0ex}}j=1,\dots ,d$

b) ${V}^{-1}{e}_{j}={\sum }_{i=1}^{h}{\alpha }_{\mathrm{ij}}{e}_{i},\phantom{\rule{thickmathspace}{0ex}}j=1,\dots ,h$

Let $\left({\beta }_{\mathrm{kl}}\right)$ denote the inverse matrix of $\left({\alpha }_{\mathrm{ij}}\right)$. Let $B:=\left({w}_{0}\left({\beta }_{\mathrm{kl}}\right)\mathrm{modulo}\phantom{\rule{thickmathspace}{0ex}}p{\right)}_{k,l=d+1,\dots ,k}$ let ${B}^{\left(p\right)}$ deonte the matrix obtained by raising all entries to the $p$-th power. $\left({\alpha }_{\mathrm{ij}}\right)$ is said to satisfy the $V$-nilpotence condition if there is a natural number $N\in ℕ$ such that ${B}^{{p}^{N-1}}\cdot \dots \cdot {B}^{\left(p\right)}\cdot B=0$.

Then a $3n$-display satisfying the $V$-nilpotence condition locally on the spectrum $\mathrm{Spec}R$ is called a display.

## Properties

The following theorem compares the Dieudonné theory in terms of displays and the crystalline Dieudonné theory? of Grothendieck and Messing Messing.

###### Theorem

Let $𝒫:=\left(P,Q,F,{V}^{-1}\right)$ be a display over $R$. Then there is an isomorphism of crystals

${D}_{𝒫}\stackrel{\sim }{\to }{D}_{B{T}_{𝒫}}\left(S\right)$D_\mathcal{P}\stackrel{\sim}{\to}\mathbf{D}_{B T_\mathcal{P}}(S)

where on the right side is the crystal defined in Messing.

## References

Revised on June 5, 2012 15:59:00 by Stephan Alexander Spahn (178.195.231.138)