the Barsotti-Tate group of an abelian variety

Recall that a p-divisible group $G$ has the defining properties that $p\phantom{\rule{thinmathspace}{0ex}}{\mathrm{id}}_{G}:G\to G$ is an epimorphism with finite kernel satisfying $G={\cup}_{j}\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}{p}^{j}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{id}}_{G}$.

Now let $A$ be any commutative algebraic $k$-group such that $p\phantom{\rule{thinmathspace}{0ex}}{\mathrm{id}}_{A}:A\to A$ is an epimorphism. Then

$$A(p):={\cup}_{j}\mathrm{ker}\phantom{\rule{thinmathspace}{0ex}}{p}^{j}\phantom{\rule{thinmathspace}{0ex}}{\mathrm{id}}_{A}$$

is a $p$-divisible group.

Revised on June 9, 2012 14:29:18
by Stephan Alexander Spahn
(79.227.138.186)