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Let $k$ be a perfect field of prime characteristic $p$.
Let $\underline W$ denote the codirected system of affine commutative unipotent? $Acu_k$ $k$-groups
The Witt ring $W(k)$ operates on $\underline W$ as follows:
Let $\sigma$ denote the Frobenius morphism $W(k)\to W(k)$, let $a\mapsto a^{(p^n)}$. This Frobenius is bijective since $k$ is perfect. Let $a\in W(k)$, let $w\in W_n(R)$, $R\in M_k$. We define
where $a^{p^{1-n} }R$ is the image of $a^{(p^{-n})}$ in $W(R)$, and $b\cdot w\in W_n(R)$, the product of $b\in W(R)$ and $w\in W_n(R)=W(R)/T^n W(R)$. By this definition $W_n(R)$ becomes a $W(k)$-module, and $T:W_n(R)\to W_{n+1}(R)$ is a homomorphism of $W(k)$-modules since we have
For any $G\in Ac u_k$ the Dieudonné module $M(G)$ of $G$ is defined to be the $W(k)$-module
or- equivalently- $M(G)=codir(Acu_k)(G,\underline W)$ where $codir(Acu_k)$ denotes the category of codirected diagrams in $Acu_k$ as described above.
is a contravariant functor from affine commutative unipotent $k$-groups to the category of $W(k)$-modules.
This construction commutes with automorphisms of $k$. In particular it commutes with the morphism $f_k:k\to k$.
For a $W(k)$-module $M$, define $M^{(p)}:=M\otimes_{W(k),\sigma}W(k)$.
As a group $M^{(p)}=M$, but the external law is $(w,m)\mapsto w^{(p^{-1})}m$.
If $f\in Acu_k(G,W_{nk})$, then $f^{(p)}:G^{(p)}\to W^{(p)}_{nk}=W_{nk}$ is a morphism and hence a map $\begin{cases}M(G)\to M(G^{(p)})\\f\mapsto f^{(p)}\end{cases}$
(…) There is an isomorphism $M(G)^{(p)}\stackrel{\sim}{\to}M(G)^{(p)}$.
The Frobenius morphism and the Verschiebung morphisminduce morphisms of $W(k)$ modules. $F:=M(F_G):M(G)^{(p)}\to M(G)$ and $V:=M(V_G):M(G)\to M(G)^{(p)}$.
The translation morphism $T:W_{nk}\to W_{(n+1)k}$ is a monomorphism and the maps $Acu_k(G,W_{nk})\to Acu_k(G,W_{(n+1)k})$ are injective.
$Acu_k(G,W_{nk})$ can be identified with a submodule of $M(G)$, namely $Acu_k(G,W_{nk})=\{m\in M(G),V^n m=0\}$ and we say that an element of $M(G)$ is killed by a power of $V$.
Let $D_k$ be the (non-commutative) ring generated by $W(k)$ and two elements $F$ and $V$ subject to the relations
Any element of $D_k$ can be written uniquely as a finite sum
If $G\in Acu_k$, then $M(G)$ has a canonical structure of a left $D_k$-module. If $K$ is a perfect extension of $k$, then there is a canonical map of $D_k$-modules
Note that $D_K\simeq W(K)\otimes_{W(k)} D_k$ and the left hand side can also be written $D_K\otimes_{D_k}M(G)$.
The functor $M$ induces an (contravariant) equivalence
between $Acu_k$ and the category of all $D_k$-modules of $V$-torsion.
For any perfect extension $K$ \of $k$ we have that
is an isomorphism. Moreover
$G$ is algebraic iff $M(G)$ is a finitely generated $D_k$-module.
$G$ is finite iff $M(G)$ is a $W(k)$-module of finite length.
Last revised on June 9, 2012 at 22:58:40. See the history of this page for a list of all contributions to it.