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p-torsion

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p-torsion of abelian groups

Since any abelian group GG is a \mathbb{Z}-module we can form for any zz\in \mathbb{N} the torsion subgroup

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

Of particular interest in this article are those cases where z=p nz=p^n for a prime number pp and a natural number nn.

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

S p(G):=colim nG[p n]S_p(G):=colim_n G[p^n]

and

T p(G):=limG[p n]T_p(G):=lim G[p^n]

Here S p(G)S_p(G) sometimes is called pp-torsion subgroup; if GG is finite S p(G)S_p(G) is also called Sylow p-subgroup of GG.

T p(G)T_p(G) is called p-adic Tate module of GG.

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

G[p]=(ker(gg n))G[p]=(ker\, (g\mapsto g^n))

In this form we can extend the Frobenius and hence this notion of pp-torsion from abelian groups to fields if we require our field to be of characteristic pp such that we have (a+b) n=a n+b n(a+b)^n=a^n+b^n.

In fact the definition of pp-torsion via the Frobenius has the advantage that we get additionally an adjoint notion to pp-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 XX denotes some scheme over a kk-ring for kk being a field of characteristic pp, we define its pp-torsion component-wise by X (p)(R):=X(s *R)X^{(p)}(R):=X(s_* R).

p-torsion of group schemes

Example

(the pp-adic Tate module)

Let GG be a commutative group scheme over a field kk with separable closure k sepk^{sep}.

Then T p(G(k sep))T_p(G(k^{sep})) is called the pp-adic Tate module of GG.

This Tate module enters the Tate conjecture?.

If GG is an abelian variety T p(G(k sep))T_p(G(k^{sep})) is equivalently the first homology group of GG.

p-divisible groups

(main article: p-divisible group)

Sometimes the information encoded in the colimit T p(G)=colim nG[p n]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

0G[p]pG[p 2]ppG[p n]pG[p n+1]p0\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 pp-divisible group of GG. Here pp denotes the multiplication-with-pp map.

We have

(1) The G[p i]G[p^i] are finite group schemes.

(2) The sequences of the form

0kerp jι jkerp j+kp jkerp k00\to ker\, p^j\xhookrightarrow{\iota_j} ker p^{j+k}\stackrel{p^j}{\to}ker p^k\to 0

are exact.

(3) G= jkerp jid GG=\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 cardkerp=p hcard \ker \,p=p^h for some natural number hh. By pars pro toto we call p hp^h also the rank of the whole sequence and hh we call its height.

Conversely we can define a pp-divisible group to be a codirected diagram

G 1i 1G 2i 2G_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)