## p-torsion of abelian groups Since any abelian group $G$ is a $\mathbb{Z}$-module we can form for any $z\in \mathbb{N}$ the set $$G[z]:=\{z|z\in G, g^z=0\}$$ which is a subgroup of $G$ called *$z$-[[nLab:torsion]] subgroup of $G$*. Of particular interest are those cases where $z=p^n$ for a [[nLab:prime number]] and a natural number $n$. There are two important constructions to perform with these $G[p^n]$ namely taking limits and colimits: $$S_p(G):=colim_n G[p^n]$$ and $$T_p(G):=lim G[p^n]$$ Here $S_p(G)$ sometimes itself is called *$p$-torsion subgroup*; if $G$ is finite $S_p(G)$ is also called *[[Sylow p-subgroup]] of $G$*. $T_p(G)$ is called [[nLab:Tate module|p-adic Tate module]] of $G$. ## p-torsion of fields $G[p]$ is obviously the kernel of the [[nLab:Frobenius]] endomorphism of $G$: $$G[p]=(ker\, (g\mapsto g^n))$$ 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 *Verschiebung*; this is explained at [[nLab:Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism ]]. ## p-torsion 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 $X^{(p)}(R):=X(R^{(p)})$. ## p-torsion of group schemes +-- {: .num_example} ###### Example (*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 *the $p$-adic Tate module of $G$*. =-- 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 $G$. ## p-divisible groups Sometimes the information encoded in the limit $T_p(G)=lim_n G[p^n]$ is considered to be not sufficient and one want instead study the directed system $$G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots$$ itself.