Barsotti–Tate group

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title: "Barsotti–Tate group" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["algebraic-groups"] topic_path: "general/algebraic-groups" source: "https://en.wikipedia.org/wiki/Barsotti–Tate_group" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

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**nth 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 2d where d is the dimension of the Abelian variety.

References

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