## 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 $$G[z]:=\{z|z\in G, zg=0\}$$ Of particular interest in this article are those cases where $z=p^n$ for a [[nLab: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: $$S_p(G):=colim_n G[p^n]$$ and $$T_p(G):=lim G[p^n]$$ Here $S_p(G)$ sometimes is called *$p$-[[nLab: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 $G$. Note that sometimes by "the Tate module" is meant a specific example of a Tate module. This example is mentioned below. ## 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 (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 $$0\to G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots$$ itself. This system is called *$p$-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 $$0\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0$$ 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 *height*. Conversely we can define a $p$-divisible group to be a codirected diagram $$G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}\dots$$ satisfying (1)(2)(3). ## References * see the references at [[nLab:p-divisible group]], in particular the notes by Richard Pink.