Spahn p-torsion (Rev #1, changes)

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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 set

G[z]:={z|zG,g z=0}G[z]:=\{z|z\in G, g^z=0\}

which is a subgroup of GG called zz-torsion subgroup of GG.

Of particular interest are those cases where z=p nz=p^n for a prime number 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 itself 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.

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(R (p))X^{(p)}(R):=X(R^{(p)}).

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.

Revision on July 18, 2012 at 16:45:35 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.