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\begin{document}

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\section*{p-torsion}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{group_theory}{}\paragraph*{{Group Theory}}\label{group_theory}

[[!include group theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{ptorsion_of_abelian_groups}{p-torsion of abelian groups}\dotfill \pageref*{ptorsion_of_abelian_groups} \linebreak
\noindent\hyperlink{ptorsion_of_fields}{p-torsion of fields}\dotfill \pageref*{ptorsion_of_fields} \linebreak
\noindent\hyperlink{ptorsion_of_schemes}{p-torsion of schemes}\dotfill \pageref*{ptorsion_of_schemes} \linebreak
\noindent\hyperlink{ptorsion_of_group_schemes}{p-torsion of group schemes}\dotfill \pageref*{ptorsion_of_group_schemes} \linebreak
\noindent\hyperlink{pdivisible_groups}{p-divisible groups}\dotfill \pageref*{pdivisible_groups} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{ptorsion_of_abelian_groups}{}\subsection*{{p-torsion of abelian groups}}\label{ptorsion_of_abelian_groups}

Since any [[abelian group]] $G$ is a $\mathbb{Z}$-module we can form for any $z\in \mathbb{N}$ the [[torsion subgroup]]

\begin{displaymath}
G[z]:=\{g|g\in G,  z g = 0\}
\end{displaymath}
Of particular interest in this article are those cases where $z=p^n$ for a [[prime number]] $p$ and a [[natural number]] $n$.

There are two important constructions to perform with these $G[p^n]$ namely taking [[limits]] and [[colimits]]:

\begin{displaymath}
S_p(G):=colim_n G[p^n]
\end{displaymath}
and

\begin{displaymath}
T_p(G):=lim G[p^n]
\end{displaymath}
Here $S_p(G)$ sometimes is called \emph{$p$-[[torsion subgroup]]}; if $G$ is finite $S_p(G)$ is also called \emph{[[Sylow p-subgroup]] of $G$}.

$T_p(G)$ is called [[nLab:Tate module|p-adic Tate module]] of $G$.

Note that sometimes by ``the Tate module'' is meant a specific example of a Tate module. This example is mentioned below.

\hypertarget{ptorsion_of_fields}{}\subsection*{{p-torsion of fields}}\label{ptorsion_of_fields}

$G[p]$ is obviously the kernel of the [[nLab:Frobenius]] endomorphism of $G$:

\begin{displaymath}
G[p]=(ker\, (g\mapsto g^n))
\end{displaymath}
In this form we can extend the Frobenius and hence this notion of $p$-torsion from abelian groups to fields if we require our field to be of characteristic $p$ such that we have $(a+b)^n=a^n+b^n$.

In fact the definition of $p$-torsion via the Frobenius has the advantage that we get additionally an adjoint notion to $p$-torsion which is sometimes called \emph{Verschiebung}; this is explained at [[nLab:Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism ]].

\hypertarget{ptorsion_of_schemes}{}\subsection*{{p-torsion of schemes}}\label{ptorsion_of_schemes}

If $X$ denotes some [[nLab:scheme]] over a $k$-ring for $k$ being a field of characteristic $p$, we define its $p$-torsion component-wise by $X^{(p)}(R):=X(s_* R)$.

\hypertarget{ptorsion_of_group_schemes}{}\subsection*{{p-torsion of group schemes}}\label{ptorsion_of_group_schemes}

\begin{example}
\label{}\hypertarget{}{}
(\emph{the} $p$-adic [[nLab:Tate module]])

Let $G$ be a commutative group scheme over a field $k$ with separable closure $k^{sep}$.

Then $T_p(G(k^{sep}))$ is called \emph{the $p$-adic Tate module of $G$}.

\end{example}
This Tate module enters the [[Tate conjecture]].

If $G$ is an abelian variety $T_p(G(k^{sep}))$ is equivalently the first homology group of $G$.

\hypertarget{pdivisible_groups}{}\subsection*{{p-divisible groups}}\label{pdivisible_groups}

(main article: [[nLab:p-divisible group]])

Sometimes the information encoded in the colimit $T_p(G)=colim_n G[p^n]$ (we passed a contravariant functor from rings to schemes) is considered to be not sufficient and one wants more generally to study the codirected system

\begin{displaymath}
0\to G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots
\end{displaymath}
itself. This system is called \emph{$p$-divisible group of $G$}. Here $p$ denotes the multiplication-with-$p$ map.

We have

(1) The $G[p^i]$ are finite group schemes.

(2) The sequences of the form

\begin{displaymath}
0\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0
\end{displaymath}
are exact.

(3) $G=\cup_j ker\, p^j\cdot id_G$

We have as cardinality (in group theory also called ``rank'') of the first item of the sequence $card \ker \,p=p^h$ for some natural number $h$. By pars pro toto we call $p^h$ also the rank of the whole sequence and $h$ we call its \emph{height}.

Conversely we can define a $p$-divisible group to be a codirected diagram

\begin{displaymath}
G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}\dots
\end{displaymath}
satisfying (1)(2)(3).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[torsion approximation]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item see the references at [[nLab:p-divisible group]], in particular the notes by Richard Pink.

\end{itemize}


\end{document}