Spahn p-torsion (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

G[z]:=\{z|z\in G, zg=0\}

Of particular interest ~~are~~ in this article are those cases where $z=p^n$ for a prime number ~~and~~ a ~~natural~~ ~~number~~ $n p$ n 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 $p$ -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 $G$ :

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 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

Example

(the $p$ -adic Tate module)

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

Then $T_p(G(k^{sep}))$ 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: 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 p-divisible group, in particular the notes by Richard Pink.

Last revised on July 18, 2012 at 22:45:48. See the history of this page for a list of all contributions to it.