is called p-adic Tate module of .
Note that sometimes by “the Tate module” is meant a specific example of a Tate module. This example is mentioned below.
is obviously the kernel of the Frobenius endomorphism of :
In this form we can extend the Frobenius and hence this notion of -torsion from abelian groups to fields if we require our field to be of characteristic such that we have .
In fact the definition of -torsion via the Frobenius has the advantage that we get additionally an adjoint notion to -torsion which is sometimes called Verschiebung; this is explained at Demazure, lectures on p-divisible groups, I.9, the Frobenius morphism.
If denotes some scheme over a -ring for being a field of characteristic , we define its -torsion component-wise by .
(the -adic Tate module)
Let be a commutative group scheme over a field with separable closure .
Then is called the -adic Tate module of .
This Tate module enters the Tate conjecture.
If is an abelian variety is equivalently the first homology group of .
(main article: p-divisible group)
Sometimes the information encoded in the colimit (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
itself. This system is called -divisible group of . Here denotes the multiplication-with- map.
(1) The are finite group schemes.
(2) The sequences of the form
We have as cardinality (in group theory also called “rank”) of the first item of the sequence for some natural number . By pars pro toto we call also the rank of the whole sequence and we call its height.
Conversely we can define a -divisible group to be a codirected diagram