In great generality, for an integer a -divisible group is a codirected diagram of abelian group objects in a category where the abelian-group objects are (equivalently) the kernels of the map given by multiplication with a power of ; these kernels are also called -torsions.
In the classically studied case is a prime number, is the category of schemes over a commutative ring (mostly a field with prime characteristic) and the abelian group schemes occurring in the diagram are assumed to be finite. In this case the diagram defining the -divisible group can be described in terms of the growth of the order of the group schemes in the diagram.
Note that there is also a notion of divisible group.
The eponymous (-divisible groups are sometimes called Barsotti-Tate groups) example is a special case of the previous one - namely the Barsotti-Tate group of an abelian variety. Let be an abelian variety over of dimension , then the multiplication map by has kernel which is a finite group scheme over of order . The natural inclusions satisfy the conditions for the limit denoted to be a -divisible group of height .
A theorem of Serre and Tate says that there is an equivalence of categories between divisible, commutative, formal Lie groups over and the category of connected -divisible groups over given by , where . In particular, every connected -divisible group is smooth
Given a -divisible group , each individual has a Cartier dual since they are all group schemes. There are also maps that make the composite the multiplication by on . After taking duals, the composite is still the multiplication by map on , so it is easily checked that forms a -divisible group called the Cartier dual.
One of the important properties of the Cartier dual is that one can determine the height of a -divisible group (often a hard task when in the abstract) using the information of the dimension of the formal group and its dual. For any -divisible group, , we have the formula that .
For the moment see display of a p-divisible group.
The dual .
For an abelian variety , the dual is where denotes the dual abelian variety. Another proof that has height is to note that and have the same dimension , so using our formula for height we get .
The category of étale -divisible groups is equivalent to the category of -adic representations of the fundamental group of the base scheme .
Important tools in the study of -divisible groups are Witt rings, Dieudonné modules and more generally Dieudonné theories? assigning to a -divisible group an object of linear algebra such as a display of a p-divisible group.
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