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Recall from relation of certain classes of k-groups? the following:
$Feu_k$ denotes the category of formal étale unipotent affine $k$-groups.
$Fiu_k$ denotes the category of formal infinitesimal unipotent $k$-groups.
$W(k)$ denotes the Witt ring over $k$.
For $D_k$ see D_k-module? in III.5, Dieudonné modules (affine unipotent groups).
is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules.
Recall moreover from III.5, Dieudonné modules (affine unipotent groups) that $Tor_V D_k-Mod:=Acu_k(G,W_{nk})=\{m\in M(G)|V^n m =0\}$ is a submodule.
which are $W(k)$-modules of finite length, killed by a power of $V$, Definition Verschiebung morphism, and on which $F$, Definition Frobenius morphism, is bijective (resp. and killed by a power of $F$).
(formulation of the statement is unclear) The functor
which is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules induces the following contravariant equivalences of categories:
$Feu_k\to Tor_V D_k-Mod\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $V$ on which $F$ is bijective.
$Fiu_k\to Tor_F D_k-Mod\hookrightarrow M(G)$ between the category of affine étale unipotent $k$-groups to the category of $W_k$-modules of finite length, killed by a power of $F$ (and killed by a power of $V$ ?)on which $F$ is bijective.
This follows from the theorem, and the fact that if $G$ is finite, then G is étale (resp, infinitesimal) if and only if $F_G$ is an isomorphism (resp. $F_G^n = 0$ for large $n$).
If $G=(\mathbb{Z}/p\mathbb{Z})_k\in Feu_k$, then $M(G)=k$ with $F=1$, $V=0$.
If $G=p \alpha_k\in Fiu_k$, then $M(G)=k$ with $F=0$, $V=0$.
For $G\in Feu_k$ or $G\in Fiu_k$, we have
rk(G)= p^{length(M(G))
Let $m,n$ be two positive integers. Then
which is bijective.
There is an isomorphism
In prose this means that the autoduality $G\mapsto D(G)$ of $Fiu_k$ corresponds via the Dieudonné-functor $M$ to the autoduality $M\to M^*$ in the category $fin Tor_{V,F}D_k-Mod$ of $D_k$-modules of finite length killed by a power of $V$ and $F$.
(Dieudonné-module of an infinitesimal multiplicative $k$-group)
Let $G\in Fim_k$. Then the Dieudonné-module of $M(G)*$ is defined by
It follows by the Cartier duality between $Fim_k$ and $Feu_k$ that the functor $G\mapsto M(G)$ induces a contravariant equivalence
between $Fim_k$ and the category of all $D_k$-modules of finite length on which $F$ is nilpotent and $V$ is bijective.
Let $G\in Fimd_k$ (i.e. $G\in Fim_k$ and $G$ diagonalizable).
$G=D(\Gamma_k)$. Then $D(G)\simeq \Gamma_k$, and
$M(D(G))=colim Acu_k(\Gamma_k,W_{nk})=colimGr(\Gamma,W_n(k))=Gr(\Gamma,W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z},W_\infty)=Mod_{W(k)}(W(k)\otimes_\mathbb{Z} \Gamma, W_\infty)$
and hence
where $W_\infty=Quot(W(k))/W(k)=colim_n W_n(k)=\underline W(k)$, see p.66.
For $F$ and $V$ we have
a) The Dieudonné functor
is a contravariant equivalence between all finite $k$-groups of $p$-torsion, and the category of all triples $(M,F_M,V_M)$ where $M$ is a finite length $W(k)$-module and $F_M$, $V_M$ two group endomorphisms of $M$ satisfying
b) $G$ is étale, infinitesimal, unipotent or multiplicative according as $F_M$ is isomorphic, $F_M$ is nilpotent, $V_M$ is nilpotent, or $V_M$ is isomorphic
c) For any $G\in Fp_k$ we have $rk(G)=p^{length M(G)}$.
d) If $k$ is a perfect extension of $k$, there exists a functorial isomorphism
Michel Demazure, lectures on p-divisible groups web
Last revised on July 21, 2012 at 23:05:14. See the history of this page for a list of all contributions to it.