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.
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 . This fact is generalized to include the case of an arbitrary base scheme on which the homomorphism induced by multiplication by is locally nilpotent . 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 .
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 .
|||I. Barsotti, "Analytic methods for abelian varieties in positive characteristic" , Coll. Théorie des Groupes Algébriques (Bruxelles, 1962) , Gauthier-Villars (1962) pp. 77–85|
|||A. Grothendieck, "Groupes de Barsotti–Tate et cristaux" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 431–436|
|||B. Mazur, W. Messing, "Universal extensions and one-dimensional crystalline cohomology" , Springer (1974)|
|||W. Messing, "The crystals associated to Barsotti–Tate groups: with applications to Abelian schemes" , Springer (1972)|
|||J.-P. Serre, "Groupes -divisibles (d'après J. Tate)" , Sem. Bourbaki , 318 , Benjamin (1968)|
|||J.T. Tate, "-divisible groups" T.A. Springer (ed.) et al. (ed.) , Proc. Conf. local fields (Driebergen, 1966) , Springer (1967) pp. 158–183|
|[a1]||M. Demazure, "Lectures on -divisible groups" , Lect. notes in math. , 302 , Springer (1972)|
P-divisible group. I.V. Dolgachev (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=P-divisible_group&oldid=13472