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Demazure, lectures on p-divisible groups, III.6, Dieudonné modules (p-torsion finite k-groups)

This entry is about a section of the text

Recall from relation of certain classes of k-groups? the following:

  1. Feu kFeu_k denotes the category of formal étale unipotent affine kk-groups.

  2. Fiu kFiu_k denotes the category of formal infinitesimal unipotent kk-groups.

  3. W(k)W(k) denotes the Witt ring over kk.

  4. For D kD_k see D_k-module? in III.5, Dieudonné modules (affine unipotent groups).

  5. M:{Acu k W(k)Mod G M(G)M:\begin{cases} Acu_k&\to& W(k)-Mod \\ G&\mapsto&M(G) \end{cases}

is a contravariant functor from affine commutative unipotent kk-groups to the category of W(k)W(k)-modules.

Recall moreover from III.5, Dieudonné modules (affine unipotent groups) that Tor VD kMod:=Acu k(G,W nk)={mM(G)V nm=0}Tor_V D_k-Mod:=Acu_k(G,W_{nk})=\{m\in M(G)|V^n m =0\} is a submodule.

which are W(k)W(k)-modules of finite length, killed by a power of VV, Definition Verschiebung morphism?, and on which FF, Definition Frobenius morphism, is bijective (resp. and killed by a power of FF).

Proposition

(formulation of the statement is unclear) The functor

M:{Acu k W(k)Mod G M(G)M:\begin{cases} Acu_k&\to& W(k)-Mod \\ G&\mapsto&M(G) \end{cases}

which is a contravariant functor from affine commutative unipotent kk-groups to the category of W(k)W(k)-modules induces the following contravariant equivalences of categories:

  1. Feu kTor VD kModM(G)Feu_k\to Tor_V D_k-Mod\hookrightarrow M(G) between the category of affine étale unipotent kk-groups to the category of W kW_k-modules of finite length, killed by a power of VV on which FF is bijective.

  2. Fiu kTor FD kModM(G)Fiu_k\to Tor_F D_k-Mod\hookrightarrow M(G) between the category of affine étale unipotent kk-groups to the category of W kW_k-modules of finite length, killed by a power of FF (and killed by a power of VV ?)on which FF is bijective.

(Demazure p.69)

Proof

This follows from the theorem, and the fact that if GG is finite, then G is étale (resp, infinitesimal) if and only if F GF_G is an isomorphism (resp. F G n=0F_G^n = 0 for large nn).

Example
  1. If G=(/p) kFeu kG=(\mathbb{Z}/p\mathbb{Z})_k\in Feu_k, then M(G)=kM(G)=k with F=1F=1, V=0V=0.

  2. If G=pα kFiu kG=p \alpha_k\in Fiu_k, then M(G)=kM(G)=k with F=0F=0, V=0V=0.

Corollary

For GFeu kG\in Feu_k or GFiu kG\in Fiu_k, we have

rk(G)= p^{length(M(G))
Proposition

Let m,nm,n be two positive integers. Then

  1. The canonical injection m W nW nm^{W_n}\to W_n defines an element uM(m W n)u\in M(m^{W_n}) satisfying V nu=F n=0V^n u= F^n= 0. This gives a map
λ n,m:D k/(D kF m+DV n)M(m n W)\lambda_{n,m}: D_k/(D_k F^m + D V^n)\to M(m^W_n)

which is bijective.

Theorem

There is an isomorphism

M(D(G))M(G) *M(D(G))\to M(G)^*

In prose this means that the autoduality GD(G)G\mapsto D(G) of Fiu kFiu_k corresponds via the Dieudonné-functor MM to the autoduality MM *M\to M^* in the category finTor V,FD kModfin Tor_{V,F}D_k-Mod of D kD_k-modules of finite length killed by a power of VV and FF.

Definition

(Dieudonné-module of an infinitesimal multiplicative kk-group)

Let GFim kG\in Fim_k. Then the Dieudonné-module of M(G)*M(G)* is defined by

M(G)=M(D(G)) *M(G)=M(D(G))^*

It follows by the Cartier duality between Fim kFim_k and Feu kFeu_k that the functor GM(G)G\mapsto M(G) induces a contravariant equivalence

Fim kfinTor FBij VD kModFim_k\to fin Tor_F Bij_V D_k-Mod

between Fim kFim_k and the category of all D kD_k-modules of finite length on which FF is nilpotent and VV is bijective.

Remark

Let GFimd kG\in Fimd_k (i.e. GFim kG\in Fim_k and GG diagonalizable).

G=D(Γ k)G=D(\Gamma_k). Then D(G)Γ kD(G)\simeq \Gamma_k, and

M(D(G))=colimAcu k(Γ k,W nk)=colimGr(Γ,W n(k))=Gr(Γ,W )=Mod W(k)(W(k) ,W )=Mod W(k)(W(k) Γ,W )M(D(G))=colim Acu_k(\Gamma_k,W_{nk})=colimGr(\Gamma,W_n(k))=Gr(\Gamma,W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z},W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z} \Gamma, W_\infty)

and hence

M(G)(W(k¯) Γ) πM(G)\simeq(W(\overline k)\otimes_\mathbb{Z}\Gamma)^\pi

where W =Quot(W(k))/W(k)=colim nW n(k)=W̲(k)W_\infty=Quot(W(k))/W(k)=colim_n W_n(k)=\underline W(k), see p.66.

For FF and VV we have

F(λχ)=λ (p)pχF(\lambda \otimes \chi)=\lambda^{(p)}\otimes p\chi
V(λχ)=λ (p 1)pχV(\lambda \otimes \chi)=\lambda^{(p^{-1})}\otimes p\chi
Theorem

a) The Dieudonné functor

{F pk=Fiu k×Feu k×Fim k(finW(k)Mod,F,V) GM(G)\begin{cases} F_p_k=Fiu_k\times Feu_k\times Fim_k\to (fin W(k)-Mod,F,V) \\ G\to M(G) \end{cases}

is a contravariant equivalence between all finite kk-groups of pp-torsion, and the category of all triples (M,F M,V M)(M,F_M,V_M) where MM is a finite length W(k)W(k)-module and F MF_M, V MV_M two group endomorphisms of MM satisfying

F M(λm)=λ (p)F M(m)F_M(\lambda m)=\lambda^{(p)} F_M(m)
V M(λ (p)m)=λV M(m)V_M(\lambda^{(p)}m)=\lambda V_M(m)
F MV M=V MF M=pid MF_M V_M=V_M F_M=p\cdot id_M

b) GG is étale, infinitesimal, unipotent or multiplicative according as F MF_M is isomorphic, F MF_M is nilpotent, V MV_M is nilpotent, or V MV_M is isomorphic

c) For any GFp kG\in Fp_k we have rk(G)=p lengthM(G)rk(G)=p^{length M(G)}.

d) If kk is a perfect extension of kk, there exists a functorial isomorphism

M(D(G))=M(G) *M(D(G))=M(G)^*

References

Michel Demazure, lectures on p-divisible groups web

Revised on July 21, 2012 23:05:14 by Stephan Alexander Spahn (79.227.141.42)