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Demazure, lectures on p-divisible groups, III.8, Dieudonné modules (p-divisible groups)

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Lemma

Let M n+1π nM nπ n1π nM 1M_{n+1}\stackrel{\pi_n}{\to}M_n\stackrel{\pi_{n-1}}{\to}\cdots\stackrel{\pi_n}{\to}M_1 a system of W(k)W(k)-modules? such that for all nn

  1. M n+1p nM n+1π nM n0M_{n+1}\stackrel{p^n}{\to}M_{n+1}\stackrel{\pi_n}{\to}M_n\to 0 is exact

  2. M nM_n is of finite length.

then M:=limM nM:=lim M_n is a finitely generated W(k)W(k)-module and the canonical map MM nM\to M_n identifies M nM/p nMM_n\simeq M/p^n M

Definition

(pp-torsion formal group) A formal group GG is called pp-torsion formal group if

  1. G=kerp nid GG=\cup ker p^n id_G

  2. kerpid Gker p id_G is finite.

There are exact sequences

0kerp nkerp n+1p nkerp n+10\to ker p^n \to ker p^{n+1}\stackrel{p^n}{\to}ker p^{n+1}
0kerp nkerp n+mp nkerp m0\to ker p^n \to ker p^{n+m}\stackrel{p^n}{\to}ker p^m

showing by induction the also kerp nker p^n is finite for all nn. Define M(G)=colimM(kerp n)M(G)= colim M(ker p^n)

Theorem

GM(G)G\to M(G) is a (contravariant) equivalence between the category of pp-torsion?formal groups and the category of tuples (M,F M,V M)(M,F_M,V_M) where M is a finitely generated W(k)W(k)-module and F MF_M, V MV_M to groups of endomorphisms of MM with

F M(wm)=w (p)F M(m)F_M(wm)=w^{(p)}F_M (m)
V M(w m (p))=wV n(m)V_M(w^{(p)}_m)=w V_n(m)
F MV M=V MF M=pid MF_M V_M=V_M F_M=p id_M

It follows from the lemma that M(G)M(G) is finitely generated and that

M n=M(G)/p nM(G)M_n=M(G)/p^n M(G)

Conversely if MM is as before we define G:=colimG nG:=colim G_n where M(G n)=M/p nMM(G_n)=M/p^n M

Moreover we have:

  1. GG is finite iff M(G)M(G) is finite and in that case M(G)M(G) is the same as in § 7.

  2. GG is pp-divisible iff M(G)M(G) is torsion-less (= free) and height(G)=dimM(G)height(G)=dim M(G).

  3. For any perfect extension? K/kK/k there is a functorial isomorphism M(G kK)W(k) W(k)M(G)M(G\otimes_k K)\simeq W(k)\otimes_{W(k)}M(G)

  4. If GG is pp-divisible with Serre dual? G G^\prime then M(G )=Mod W(k)(M(G),W(k)M(G^\prime)=Mod_{W(k)}(M(G),W(k) with

(F M(G )f)(m)=f(V Mm)(p)(F_{M(G^\prime)}f)(m)=f(V_M m)(p)

and

(V M(G )f)(m)=f(F Mm) (p 1)(V_{M(G^\prime)}f)(m)=f(F_M m)^{(p^{-1})}

(Demazure Theorem p.71-72)

Theorem

a) The Dieudonné functor

{Torf p(finW(k)Mod,F,V) GM(G)\begin{cases} Torf_p\to (fin W(k)-Mod,F,V) \\ G\to M(G) \end{cases}

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

F M(λm)=λ pF 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

It follows from the lemma that M(G)M(G) is finitely generated and M nM(G)/p nM(G)M_n\simeq M(G)/p^n M(G). Conversely if MM is as before, then we define GG as colimG ncolim G_n where M(G n)=M/p nMM(G_n)= M/ p^n M.

Remark

From the definition and what we already verified follows:

  1. GG is finite iff M(G)M(G) is finite, and in that case M(G)M(G) is the same as defined in § 7.

  2. GG is finite iff M(G)M(G) is torsion-less (= free), and height(G)=dimM(G)height(G)=dim M(G).

  3. For any perfect extension K/kK/k, there is a functorial isomorphism M(G kK)W(K) W(k)M(G)M(G\otimes_k K)\simeq W(K)\otimes_{W(k)} M(G).

  4. If GG is pp-divisible, with Serre dual G G^\prime, then M(G )=Mod W(k)M(G)M(G^\prime)=Mod_{W(k)}M(G), with F M(G )f)(m)=f(V Mm) (p)F_{M(G^\prime)} f)(m)=f(V_M m)^{(p)} and (V M(G )f)(m)=f(F Mm) (p (1)(V_{M(G^\prime)}f)(m)=f(F_M m)^{(p^{(-1)}}.

References

Michel Demazure, lectures on p-divisible groups web

Revised on June 9, 2012 15:57:34 by Stephan Alexander Spahn (79.227.138.186)