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display of a p-divisible group

Idea

In general an assignation of an object of linear algebra to a pp-divisible group is called a Dieudonné theory?.

There is a Dieudonné theory assigning to a formal pp-divisible group XX over an excellent p-adic ring? RR an object called a display. On the display one can read off the structural equations for the Cartier module of XX.

Definition

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

w n:{W(R)R (x 0,,x i,)x 0 p n+px 1 p n1++p nx nw_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:{W(R)W(R) (x 0,,x i,)(0,x 0,,x i,)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 pp be a prime number and let

F:W(R)W(R)F:W(R)\to W(R)

denote the Frobenius morphism. Then Frobenius, Verschiebung, and the Witt-polynomial morphism satisfy the ‘’pp-adic Witt-Frobenius identities’’:

w n(F(x))=w n+1(x),n0w_n(F(x))=w_{n+1}(x),\; n\ge 0
w n(V(x))=w n1(x),n>0w_n(V(x))=w_{n-1}(x),\; n\gt 0
w 0(V(x))=0w_0(V(x))=0
FV=pFV=p
VF(xy)=xV(y)VF(xy)=xV(y)
Definition

A 3n3n-display over R is defined to be a quadruple (P,Q,F,V 1)(P, Q, F, V^{-1}) where PP is a finitely generated projective W(R)W(R)-module, QPQ \subset P is a submodule and FF and V 1V^{-1} are FF-linear maps F:PPF : P \to P, V 1:QPV^{-1}: Q \to P.

The following properties are satisfied:

  1. ker(w 0)PQPker(w_0)P \subset Q \subset P and P/QP /Q is a direct summand of the W(R)W (R)−module P/ker(w 0)PP /ker(w_0)P.

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

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

Definition

Let (α ij)(\alpha_{ij}) be a invertible matrix satisfying

a) Fe j= i=1 hα ije i,j=1,,dF e_j=\sum_{i=1}^h\alpha_{ij} e_i, \; j=1,\dots ,d

b) V 1e j= i=1 hα ije i,j=1,,hV^{-1} e_j=\sum_{i=1}^h\alpha_{ij} e_i, \; j=1,\dots ,h

Let (β kl)(\beta_{kl}) denote the inverse matrix of (α ij)(\alpha_{ij}). Let B:=(w 0(β kl)modulop) k,l=d+1,,kB:=(w_0(\beta_{kl})modulo\; p)_{k,l=d+1,\dots,k} let B (p)B^{(p)} deonte the matrix obtained by raising all entries to the pp-th power. (α ij)(\alpha_{ij}) is said to satisfy the VV-nilpotence condition if there is a natural number NN\in \mathbb{N} such that B p N1B (p)B=0B^{p^{N-1}}\cdot\dots\cdot B^{(p)}\cdot B=0.

Then a 3n3n-display satisfying the VV-nilpotence condition locally on the spectrum SpecRSpec 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:=(P,Q,F,V 1)\mathcal{P}:=(P,Q,F,V^{-1}) be a display over RR. Then there is an isomorphism of crystals

D 𝒫D BT 𝒫(S)D_\mathcal{P}\stackrel{\sim}{\to}\mathbf{D}_{B T_\mathcal{P}}(S)

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

Examples

References

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