Spahn p-torsion (Rev #3, 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 $G$.

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 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

If This Tate module enters the$G$Tate conjecture? is an abelian variety $T_p(G(k^{sep}))$ is equivalently the first homology group of $G$.

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.