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 Barsotti (1962) under the name equidimensional hyperdomain and by Tate (1967) under the name p-divisible groups, and named Barsotti–Tate groups by Grothendieck (1971).

Tate (1967) 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 pⁿ and such that G_n is (identified with) the group of elements of order pⁿ in G_n+1.

More generally, Grothendieck (1971) 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 pⁿ is a scheme of rank p^nh, and G is the direct limit of these subgroups.

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