Contents

group theory

# Contents

## 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 torsion subgroup

$G[z] \,\coloneqq\, \big\{ g \,\vert\, g \in G, \, z g = 0 \big\} \,.$

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:

$S_p(G) \,\coloneqq\, colim_n G[p^n]$
$T_p(G) \,\coloneqq\, lim_n G[p^n] \,.$

Here $S_p(G)$ is sometimes called the $p$-torsion subgroup; if $G$ is finite then $S_p(G)$ is also called Sylow p-subgroup of $G$.

$T_p(G)$ 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

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

$G[p] \,=\, \big( ker \, (g \mapsto g^p) \big) \,.$

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 $(a+b)^p = a^p + b^p$.

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 $X^{(p)}(R) \coloneqq X(s_* R)$.

## p-torsion of group schemes

Let $G$ is a commutative group scheme over a scheme $S$. Define the multiplication by $p$ map as follows - $[p] \colon G \xrightarrow{\Delta} G \times_S .... \times_S G \to G$.

Because $G$ is a commutative group scheme, this is a map in the category of group schemes. The $[p]$ torsion, $G[p]$, is then the pullback along the identity section of the multiplication by $[p]$ map.

The fiber of $G[p]$ at a given $s \in S$ is a group. (Its a group scheme over the residue field of $s$). And it is the $p$-torsion in the fiber of $G$ at $s$.

The notions of $S_p$, (as an Ind-scheme) and $T_p$ (as a scheme) readily generalize using this notion of $p$-torsion.

###### Example

(the $p$-adic Tate module)

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

Then $T_p\big(G(k^{sep})\big)$ 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: 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).

see the references at p-divisible group, in particular the notes

• Richard Pink, Shatz Group Schemes, Formal Groups, and $p$-Divisible Groups