P-divisible group
Barsotti–Tate group
A generalization of the concept of a commutative formal group of finite height. The homomorphism induced by multiplication by the prime number is an epimorphism for a
-divisible group.
Let be a scheme and let
be a prime number. A
-divisible group of height
is an inductive system
of commutative finite group schemes
of order
(cf. Group scheme) such that the sequences
![]() |
are exact (cf. Exact sequence; here is the homomorphism of multiplication by
). A morphism of
-divisible groups is a morphism of the inductive systems. A
-divisible group is said to be connected (respectively, étale) if all
are connected (respectively, étale) group schemes. A connected
-divisible group over a field of characteristic
is a commutative formal group (considered as the inductive limit of the kernels of
, i.e. of multiplication by
) for which multiplication by
is an isogeny [6]. This fact is generalized to include the case of an arbitrary base scheme
on which the homomorphism induced by multiplication by
is locally nilpotent [4]. The category of étale
-divisible groups is equivalent to the category of
-adic representations of the fundamental group of the scheme
. Each
-divisible group
over an Artinian scheme
contains a maximal connected subgroup
, which is known as the connected component of the identity, the quotient by which is an étale
-divisible group. The dimension of the Lie algebra of any
is said to be the dimension of the
-divisible group
.
Let be an Abelian variety over the field
of dimension
, let
be the kernel of the homomorphism of multiplication by
in
and let
be a natural inclusion. The inductive system
is a
-divisible group of height
. Its connected component of the identity
coincides with the formal completion of
along the unit section, while the height of
equals
.
Let be a
-divisible group of height
, let
denote the Cartier dual finite group schemes and let
be the mapping dual to the mapping of multiplication by
. The system
is then a
-divisible group of height
and is called the Serre dual to the
-divisible group
. The sum of the dimensions of
and
is equal to
.
As in formal groups, so, too, in -divisible groups, the concept of a Dieudonné module, which plays an important role in the deformation theory of
-divisible groups [2], [3], [4] may be introduced.
If is the spectrum of an unequal-characteristic discrete valuation ring
with residue field of characteristic
, the structure of a
-divisible group is closely connected with the structure of the completion of the algebraic closure of the quotient field
of
, considered as a module over the Galois group of the field
[6].
References
[1] | I. Barsotti, "Analytic methods for abelian varieties in positive characteristic" , Coll. Théorie des Groupes Algébriques (Bruxelles, 1962) , Gauthier-Villars (1962) pp. 77–85 |
[2] | A. Grothendieck, "Groupes de Barsotti–Tate et cristaux" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 431–436 |
[3] | B. Mazur, W. Messing, "Universal extensions and one-dimensional crystalline cohomology" , Springer (1974) |
[4] | W. Messing, "The crystals associated to Barsotti–Tate groups: with applications to Abelian schemes" , Springer (1972) |
[5] | J.-P. Serre, "Groupes ![]() |
[6] | J.T. Tate, "![]() |
Comments
References
[a1] | M. Demazure, "Lectures on ![]() |
P-divisible group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=P-divisible_group&oldid=13472