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Let be a field of prime characteristic .
(-divisible group)
(pdg1) is an epimorphism.
(pdg2) is a -torsion group in that
(pdg3) is finite.
We have , . This is called the height of .
(alternative definition of -divisible group)
Let
be a codirected diagram of finite k-groups? such that
, a fixed integer,
all sequences are exact.
Then is a -divisible group of height and .
If is a p-divisible group in the sense of the first definition, from (pdg1) follows . Since is multiplicative
is exact.
(Serre dual? of a -divisible group)
Let be a -divisible group . The Serre dual of is defined by: let and let is the map induced by . Then we define
This is a -divisible formal group with and we have and .
Let denote the ring of p-adic integers, let denote the field of p-adic numbers. The constant formal group is a -divisible group of height .
Conversely any -divisible group of height is isomorphic to .
Let be a commutative algebraic k-group, such that is an epimorphism. Then
is finite.
is a -divisible group containing .
If we have .
If is an abelian variety of dimension is an epimorphism with and consequently is a -divisible group of height . This example is further described in chapter V, p-adic cohomology of abelian varieties?, particularly in V.3, structure of the p-divisible group A(p)?.
Let be a -formal group. Then is -divisible iff the following conditions hold:
, finite.
is of finite type, smooth, and is finite.
Let be an algebraic unipotent -group, then is never -divisible unless is finite.
Let be -divisible. Then we have
Let be a connected, finite type, smooth formal group. There exist two subgroups ,K\subseteq GHpp^n K = 0nH\cap KG=H+ K$.
Last revised on June 3, 2012 at 16:37:59. See the history of this page for a list of all contributions to it.