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'' (over a scheme ''S'') to be an inductive system of groups ''G''_''n'' for ''n''≥0, such that ''G''_''n'' is a finite group scheme over ''S'' of order ''p''^''hn'' and such that ''G''_''n'' is (identified with) the group of elements of order divisible by ''p''^''n'' in ''G''_''n''+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''(1) of order ''p'' of ''G'' are (represented by) a finite locally free scheme.
The group ''G''(1) has rank ''p''^''h'' for some locally constant function ''h'' on ''S'', called the rank or height of the group ''G''. The subgroup ''G''(''n'') of points of order ''p''^''n'' is a scheme of rank ''p''^''nh'', and ''G'' is the direct limit of these subgroups.

Example

*Take ''G''_''n'' to be the cyclic group of order ''p''^''n'' (or rather the group scheme corresponding to it). This is a ''p''-divisible group of height 1.
*Take ''G''_''n'' to be the group scheme of ''p''^''n''th roots of 1. This is a ''p''-divisible group of height 1.
*Take ''G''_''n'' to be the subgroup scheme of elements of order ''p''^''n'' of an abelian variety. This is a ''p''-divisible group of height 2''d'' where ''d'' is the dimension of the Abelian variety.

References

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{{DEFAULTSORT:Barsotti-Tate group
Algebraic groups