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\begin{document}

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\section*{Demazure, lectures on p-divisible groups, II.11, p-divisible formal groups}

[[!redirects II.11, p-divisible formal 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}
Let $k$ be a field of prime characteristic $p\gt 0$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
($p$-divisible group)

\begin{description}

\item[A commutative formal $k$-group $G$ is called \emph{[[p-divisible group|p-divisible formal k-group]]} or \emph{Barsotti-Tate group} if it satisfies the following properties:] (pdg1) $p\cdot id_G\to G$ is an epimorphism.

 
(pdg2) $G$ is a $p$-torsion group in that $G=\cup_j ker(p^j \cdot id_G)$

 
(pdg3) $ker(p\cdot id_G)$ is finite.

 
\end{description}
We have $rk(ker \,p \cdot id_G)=p^h$, $h\in \mathbb{N}$. This $h$ is called the \emph{height $ht(G)$ of $G$}.

\end{defn}
\begin{rem}
\label{}\hypertarget{}{}
(alternative definition of $p$-divisible group)

Let

\begin{displaymath}
G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}G_3\stackrel{i_3}{\to}\cdots
\end{displaymath}
be a codirected diagram of [[finite k-group|finite k-groups]] such that

\begin{enumerate}%
\item $rk(G_j)=p^{h j}$, $h$ a fixed integer,


\item all sequences $0\stackrel{}{\to}G_j\stackrel{i_j}{\to}G_{j+1}\stackrel{p^j}{\to}G^{j+1}$ are exact.



\end{enumerate}
Then $colim_n G_n$ is a $p$-divisible group of height $h$ and $ker(p^n id_G:G\to G)\simeq G_n$.

\end{rem}
\begin{remark}
\label{}\hypertarget{}{}
If $G$ is a p-divisible group in the sense of the first definition, from (pdg1) follows $rk(ker p^j \cdot id_G)0p^{j\cdot ht (G)}$. Since $rk$ is multiplicative

\begin{displaymath}
0\to ker \,p^j\hookrightarrow ker \,p^{j+k}\stackrel{p^j}{\to}ker\, p^k\to 0
\end{displaymath}
is exact.

\end{remark}
\begin{defn}
\label{}\hypertarget{}{}
([[Serre dual]] of a $p$-divisible group)

Let $G$ be a $p$-divisible group $G$. The \emph{Serre dual} $G^\prime$ of $G$ is defined by: let $G_j:=ker(p^j id_G)$ and let $p_j:G_{j+1}\to G_j$ is the map induced by $p id_G$. Then we define

\begin{displaymath}
G_j^\prime:=D(G_j)
\end{displaymath}
\begin{displaymath}
i_j^\prime:=D(p_j):G^\prime_j\to G^\prime_{j+1}
\end{displaymath}
\begin{displaymath}
G^\prime:=colim_{j^\prime}G_j^\prime
\end{displaymath}
This is a $p$-divisible formal group with $ht(G^\prime)=ht(G)$ and we have $p_j^\prime=D(i_j)$ and $(G^\prime)^\prime\simeq G$.

\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
Let $\mathbb{Z}_p$ denote the [[p-adic number|ring of p-adic integers]], let $\mathbb{Q}_p$ denote the [[p-adic number|field of p-adic numbers]]. The constant formal group $(\mathbb{Q}_p /\mathbb{Z}_p)_k$ is a $p$-divisible group of height $1$.

Conversely any $p$-divisible group of height $h$ is isomorphic to $(\mathbb{Q}_p /\mathbb{Z}_p)^h_k$.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Let $A$ be a commutative [[algebraic group|algebraic k-group]], such that $p id_G:A\to A$ is an epimorphism. Then

\begin{enumerate}%
\item $ker(p\cdot id_A)$ is finite.


\item $A(p):=\cup_j ker(p^j id_A)$ is a $p$-divisible group containing $\hat A^\circ=\cup_j ker(F^j G)$.


\item If $A=\mu_k$ we have $A(p)=\cup_j p^j \mu_k=(\mathbb{Q}_p /\mathbb{Z}_p)^\prime_k$.


\item If $A$ is an abelian variety of dimension $g$ $p id_A$ is an epimorphism with $rk(her p id_G)=p^{2g}$ and consequently $A(p)$ is a $p$-divisible group of height $2g$. This example is further described in chapter [[V, p-adic cohomology of abelian varieties]], particularly in [[V.3, structure of the p-divisible group A(p)]].



\end{enumerate}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Let $G$ be a $k$-formal group. Then $G$ is $p$-divisible iff the following conditions hold:

\begin{enumerate}%
\item $\pi_\circ(G)(\overline k)\simeq (\mathbb{Q}_p /\mathbb{Z}_p)^r$, $r$ finite.


\item $G^\circ$ is of finite type, smooth, and $ker(V:G^{\circ (p)}\to G^\circ)$ is finite.



\end{enumerate}
\end{prop}
\begin{example}
\label{}\hypertarget{}{}
Let $A$ be an algebraic unipotent $k$-group, then $\hat A^\circ$ is never $p$-divisible unless $A$ is finite.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
Let $G$ be $p$-divisible. Then we have $height(G)=dim(G)+dim(G^\prime)$

\end{remark}
\begin{prop}
\label{}\hypertarget{}{}
Let $G$ be a connected, finite type, smooth formal group. There exist two subgroups $H$,K$\backslash$subseteq G$with$H$is$p$-divisible,$p{\tt \symbol{94}}n K = 0$for large$n$,$H$\backslash$cap K$is finite, and$G=H+ K\$.

\end{prop}


\end{document}