Barsotti–Tate group
Encyclopedia
In algebraic geometry
, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety
in characteristic p. They were introduced by under the name equidimensional hyperdomain and by under the name p-divisible groups, and named Barsotti–Tate groups by .
such that the points G(1) of order p of G are (represented by) a finite locally free scheme.
The group G(1) has rank pd for some locally constant function d on S, called the rank or height of the group G. The subgroup G(n) of points of order pn is a scheme of rank pnd, and G is the direct limit of these subgroups.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, Barsotti–Tate groups or p-divisible groups are similar to the points of order a power of p on an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions...
in characteristic p. They were introduced by under the name equidimensional hyperdomain and by under the name p-divisible groups, and named Barsotti–Tate groups by .
Definition
defined a Barsotti–Tate group G over a scheme S to be a fppf sheaf of commutative groups over S that is p-divisible, p-torsion,such that the points G(1) of order p of G are (represented by) a finite locally free scheme.
The group G(1) has rank pd for some locally constant function d on S, called the rank or height of the group G. The subgroup G(n) of points of order pn is a scheme of rank pnd, and G is the direct limit of these subgroups.