Spahn p-torsion (Rev #4, changes)

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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

which is a subgroup of $G$ called $z$-torsion subgroup of $G$.

Of particular interest are those cases where $z=p^n$ for a prime number and a natural number $n$.

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

and

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 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 $G$:

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

p-torsion of schemes

If $X$ denotes some scheme over a $k$ -ring 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

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 (main the article: information encoded in the limit$T_p(G)=lim_n G[p^n]$p-divisible group ) is considered to be not sufficient and one want instead study the directed system

$$G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots$$

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

itself.

$$0\to G[p]\stackrel{p}{\to}G[p^2]\stackrel{p}{\to}\dots \stackrel{p}{\to}G[p^{n}]\stackrel{p}{\to}G[p^{n+1}]\stackrel{p}{\to}\dots$$

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).