group theory

# Contents

## p-torsion of abelian groups

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

$G\left[z\right]:=\left\{g\mid g\in G,zg=0\right\}$G[z]:=\{g|g\in G, z g = 0\}

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 $G\left[{p}^{n}\right]$ namely taking limits and colimits:

${S}_{p}\left(G\right):={\mathrm{colim}}_{n}G\left[{p}^{n}\right]$S_p(G):=colim_n G[p^n]

and

${T}_{p}\left(G\right):=\mathrm{lim}G\left[{p}^{n}\right]$T_p(G):=lim G[p^n]

Here ${S}_{p}\left(G\right)$ sometimes is called $p$-torsion subgroup; if $G$ is finite ${S}_{p}\left(G\right)$ is also called Sylow p-subgroup of $G$.

${T}_{p}\left(G\right)$ is called 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

$G\left[p\right]$ is obviously the kernel of the Frobenius endomorphism of $G$:

$G\left[p\right]=\left(\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}\left(g↦{g}^{n}\right)\right)$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 $\left(a+b{\right)}^{n}={a}^{n}+{b}^{n}$.

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 Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism.

## p-torsion of schemes

If $X$ denotes some scheme over a $k$-ring for $k$ being a field of characteristic $p$, we define its $p$-torsion component-wise by ${X}^{\left(p\right)}\left(R\right):=X\left({s}_{*}R\right)$.

## p-torsion of group schemes

###### Example

(the $p$-adic Tate module)

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

Then ${T}_{p}\left(G\left({k}^{\mathrm{sep}}\right)\right)$ is called the $p$-adic Tate module of $G$.

This Tate module enters the Tate conjecture?.

If $G$ is an abelian variety ${T}_{p}\left(G\left({k}^{\mathrm{sep}}\right)\right)$ is equivalently the first homology group of $G$.

## p-divisible groups

(main article: p-divisible group)

Sometimes the information encoded in the colimit ${T}_{p}\left(G\right)={\mathrm{colim}}_{n}G\left[{p}^{n}\right]$ (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\left[p\right]\stackrel{p}{\to }G\left[{p}^{2}\right]\stackrel{p}{\to }\dots \stackrel{p}{\to }G\left[{p}^{n}\right]\stackrel{p}{\to }G\left[{p}^{n+1}\right]\stackrel{p}{\to }\dots$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 $G\left[{p}^{i}\right]$ are finite group schemes.

(2) The sequences of the form

$0\to \mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}{p}^{j}\stackrel{{\iota }_{j}}{↪}\mathrm{ker}{p}^{j+k}\stackrel{{p}^{j}}{\to }\mathrm{ker}{p}^{k}\to 0$0\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0

are exact.

(3) $G={\cup }_{j}\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}{p}^{j}\cdot {\mathrm{id}}_{G}$

We have as cardinality (in group theory also called “rank”) of the first item of the sequence $\mathrm{card}\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}p={p}^{h}$ 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

${G}_{1}\stackrel{{i}_{1}}{\to }{G}_{2}\stackrel{{i}_{2}}{\to }\dots$G_1\stackrel{i_1}{\to}G_2\stackrel{i_2}{\to}\dots

satisfying (1)(2)(3).

## References

Revised on November 26, 2012 21:37:45 by Urs Schreiber (82.169.65.155)