Spahn display of a formal p-divisible group (Rev #3, changes)

Showing changes from revision #2 to #3: Added | ~~Removed~~ | ~~Chan~~ged

~~Let $R\in CRing$ a commutative unitary ring. Let $W(R)$ denote the ring of Witt vectors of $R$ . Let~~

Idea

~~$w_n:\begin{cases} W(R)\to R \\ (x_0,\dots,x_i,\dots)\mapsto x_0^{p^n}+p x_1^{p^{n-1}}+ \dots + p^n x_n \end{cases}$~~

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 \geq 0

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

w_{n} (V (x)) = w_{n - 1} (x), n > 0

~~w_n(V(x))=w_{n-1}(x)\;~~ 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:

$ker(w_0)P \subset Q \subset P$ and $P /Q$ is a direct summand of the $W (R)$ −module $P /ker(w_0)P$ .
$V^{-1} : Q \to P$ is a $F$ -linear epimorphism.
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

Revision on June 4, 2012 at 14:37:00 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.