nLab p-torsion



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

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) \,\coloneqq\, colim_n G[p^n]
T p(G)lim nG[p n]. T_p(G) \,\coloneqq\, lim_n G[p^n] \,.

Here S p(G)S_p(G) is sometimes called the pp-torsion subgroup; if GG is finite then 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 p)). G[p] \,=\, \big( ker \, (g \mapsto g^p) \big) \,.

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

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) \coloneqq X(s_* R).

p-torsion of group schemes

Let GG is a commutative group scheme over a scheme SS. Define the multiplication by pp map as follows - [p]:GΔG× S....× SGG[p] \colon G \xrightarrow{\Delta} G \times_S .... \times_S G \to G.

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

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

The notions of S pS_p, (as an Ind-scheme) and T pT_p (as a scheme) readily generalize using this notion of pp-torsion.


(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\big(G(k^{sep})\big) 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).


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

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

Last revised on December 15, 2021 at 18:41:01. See the history of this page for a list of all contributions to it.