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Demazure, lectures on p-divisible groups, III.5, Dieudonné modules (affine unipotent groups)

This entry is about a section of the text

Let kk be a perfect field of prime characteristic pp.

Definition

Let W̲\underline W denote the codirected system of affine commutative unipotent? Acu kAcu_k kk-groups

W 1kTW 2kTW 3kTW_{1k}\stackrel{T}{\to}W_{2k}\stackrel{T}{\to}W_{3k}\stackrel{T}{\to}\cdots

The Witt ring W(k)W(k) operates on W̲\underline W as follows:

Let σ\sigma denote the Frobenius morphism W(k)W(k)W(k)\to W(k), let aa (p n)a\mapsto a^{(p^n)}. This Frobenius is bijective since kk is perfect. Let aW(k)a\in W(k), let wW n(R)w\in W_n(R), RM kR\in M_k. We define

a*w:=a p 1nRwa * w:=a^{p^{1-n}} R\cdot w

where a p 1nRa^{p^{1-n} }R is the image of a (p n)a^{(p^{-n})} in W(R)W(R), and bwW n(R)b\cdot w\in W_n(R), the product of bW(R)b\in W(R) and wW n(R)=W(R)/T nW(R)w\in W_n(R)=W(R)/T^n W(R). By this definition W n(R)W_n(R) becomes a W(k)W(k)-module, and T:W n(R)W n+1(R)T:W_n(R)\to W_{n+1}(R) is a homomorphism of W(k)W(k)-modules since we have

T(a*w)=T(a (p 1n)Rw)=T(F(a (p n))R)w)=a p nTw=a*TwT(a* w)= T(a^{(p^{1-n)}} R\cdot w)=T(F(a^{(p^{-n})})R)\cdot w)=a^{p^{-n}}\cdot T w=a * Tw

For any GAcu kG\in Ac u_k the Dieudonné module M(G)M(G) of GG is defined to be the W(k)W(k)-module

M(G)=colim nAcu k(G,W nk)M(G)=colim_n Acu_k(G,W_{nk})

or- equivalently- M(G)=codir(Acu k)(G,W̲)M(G)=codir(Acu_k)(G,\underline W) where codir(Acu k)codir(Acu_k) denotes the category of codirected diagrams in Acu kAcu_k as described above.

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.

This construction commutes with automorphisms of kk. In particular it commutes with the morphism f k:kkf_k:k\to k.

Definition and Remark
  1. For a W(k)W(k)-module MM, define M (p):=M W(k),σW(k)M^{(p)}:=M\otimes_{W(k),\sigma}W(k).

  2. As a group M (p)=MM^{(p)}=M, but the external law is (w,m)w (p 1)m(w,m)\mapsto w^{(p^{-1})}m.

  3. If fAcu k(G,W nk)f\in Acu_k(G,W_{nk}), then f (p):G (p)W nk (p)=W nkf^{(p)}:G^{(p)}\to W^{(p)}_{nk}=W_{nk} is a morphism and hence a map {M(G)M(G (p)) ff (p)\begin{cases}M(G)\to M(G^{(p)})\\f\mapsto f^{(p)}\end{cases}

  4. (…) There is an isomorphism M(G) (p)M(G) (p)M(G)^{(p)}\stackrel{\sim}{\to}M(G)^{(p)}.

  5. The Frobenius morphism and the Verschiebung morphisminduce morphisms of W(k)W(k) modules. F:=M(F G):M(G) (p)M(G)F:=M(F_G):M(G)^{(p)}\to M(G) and V:=M(V G):M(G)M(G) (p)V:=M(V_G):M(G)\to M(G)^{(p)}.

  6. The translation morphism T:W nkW (n+1)kT:W_{nk}\to W_{(n+1)k} is a monomorphism and the maps Acu k(G,W nk)Acu k(G,W (n+1)k)Acu_k(G,W_{nk})\to Acu_k(G,W_{(n+1)k}) are injective.

  7. Acu k(G,W nk)Acu_k(G,W_{nk}) can be identified with a submodule of M(G)M(G), namely Acu k(G,W nk)={mM(G),V nm=0}Acu_k(G,W_{nk})=\{m\in M(G),V^n m=0\} and we say that an element of M(G)M(G) is killed by a power of VV.

Definition and Remark

Let D kD_k be the (non-commutative) ring generated by W(k)W(k) and two elements FF and VV subject to the relations

Fw=w (p)FFw=w^{(p)} F
w (p)V=Vww^{(p)}V=V w
FV=VF=pFV=VF=p

Any element of D kD_k can be written uniquely as a finite sum

Σ i>0a iV i+a 0+Σ i>0a iF i\Sigma_{i\gt 0}a_{-i}V^i + a_0 + \Sigma_{i\gt 0}a_i F^i

If GAcu kG\in Acu_k, then M(G)M(G) has a canonical structure of a left D kD_k-module. If KK is a perfect extension of kk, then there is a canonical map of D kD_k-modules

W(K) W(k)M(G)M(G kK)W(K)\otimes_{W(k)}M(G)\to M(G \otimes_k K)

Note that D KW(K) W(k)D kD_K\simeq W(K)\otimes_{W(k)} D_k and the left hand side can also be written D K D kM(G)D_K\otimes_{D_k}M(G).

Theorem

The functor MM induces an (contravariant) equivalence

Acu kTor VD kModAcu_k\to Tor_V D_k Mod

between Acu kAcu_k and the category of all D kD_k-modules of VV-torsion.

For any perfect extension KK \of kk we have that

W(K) W(k)M(G)M(G kK)W(K)\otimes_{W(k)}M(G)\to M(G \otimes_k K)

is an isomorphism. Moreover

  1. GG is algebraic iff M(G)M(G) is a finitely generated D kD_k-module.

  2. GG is finite iff M(G)M(G) is a W(k)W(k)-module of finite length.

Revised on June 9, 2012 22:58:40 by Stephan Alexander Spahn (79.227.170.74)