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\begin{document}

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\section*{Demazure, lectures on p-divisible groups, II.9, unipotent affine groups, decomposition of affine groups}

[[!redirects II.9, unipotent affine groups, decomposition of affine groups]] This entry is about a section of the text

\begin{itemize}%
\item Michel [[Demazure, lectures on p-divisible groups]], \href{http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups}{web}

\end{itemize}
\begin{theorem}
\label{}\hypertarget{}{}
Let $G$ be an [[affine scheme|affine]] [[group scheme|k-group]]. Then the following conditions are equivalent.

\begin{enumerate}%
\item The [[completion]] of the [[Cartier duality|Cartier dual]] $\hat D(G)$ of $G$ is a connected formal group.


\item Any [[multiplicative group scheme|multiplicative subgroup]] of $G$ is zero.


\item For any subgroup $H$ of $G$ with $H\neq 0$ we have $Gr_k(H,\alpha_k)\neq 0$.


\item Any algebraic quotient of $G$ is an extension of subgroups of $\alpha_k$.


\item (If $p\neq 0)$, $\cap Im V^n_G =e$.



\end{enumerate}
\end{theorem}
\begin{defn}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item A group satisfying the conditions of the previous theorem is called \emph{[[unipotent k-group]]}.


\item Unipotent groups correspond by duality to connected formal $k$-groups.


\item The category $ACu_k$ of affine commutative unipotent groups form a thick subcategory of $AC_k$ which is stable under limits.



\end{enumerate}
\end{defn}
The following theorem is the dual to the theorem of the previos chapter.

\begin{theorem}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item An affine $k$ group is in a unique way an extension of a unipotent group by a multiplicative group.


\item This extension splits if $k$ is perfect.


\item If $k$ is perfect any finite group is uniquely the product of four subgroups which are respectively [[étale group scheme|étale]] multiplicative, \'e{}tale unipotent, [[infinitesimal group scheme|infinitesimal]] multiplicative and infinitesimal unipotent.


\item The category $F_k$ of finite commutative $k$-groups splits as a product of four subcategories: $Fem_k$, $Feu_k$, $Fim_k$, $Fiu_k$.


\item The categories $Feu_k$ and $Fim_k$ are dual to each other.


\item The categories $Fem_k$ and $Fiu_k$ are selfdual.



\end{enumerate}
\end{theorem}
\begin{prop}
\label{}\hypertarget{}{}
\begin{enumerate}%
\item Let $p = 0$, then Then $F_k=Fem_k$.


\item Let $p\neq 0$, let $k$ be algebraically closed. Then any commutative finite $k$-group is an extension of copies of $p \alpha_k$, $p \mu_k$ and $(\mathbb{Z}/r\mathbb{Z})_k$ where $r$ is prime.



\end{enumerate}
\end{prop}
\begin{cor}
\label{}\hypertarget{}{}
If $m$ is a prime and $G$is a finite commutative $k$-group, then$m^\alpha id_G=0$ for large $\alpha$ iff $rk(G)$ is a power of $m$.

\end{cor}


\end{document}