Barsotti–Tate group


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.

Definition

defined a p-divisible group of height h to be an inductive system of groups Gn for n≥0, such that Gn is a finite group scheme over S of order pn and such that Gn is the group of elements of order divisible by pn in Gn+1.
More generally, defined a Barsotti–Tate group G over a scheme S to be an fppf sheaf of commutative groups over S that is p-divisible, p-torsion,
such that the points G of order p of G are a finite locally free scheme.
The group G has rank ph for some locally constant function h on S, called the rank or height of the group G. The subgroup G of points of order pn is a scheme of rank pnh, and G is the direct limit of these subgroups.

Example