# G is p-divisible, about the affine rings of G[p]

Let $R$ be a ring and $G$ a $p$-divisible group over $R$. Since $G[p]$ is finite flat over $\text{Spec}(R)$, it is an affine (group) scheme, say $\text{Spec}(B)$. What can be said about $B$ beyond the fact that it is a locally free $R$-algebra? Similar questions can be asked for $G[p^n]$. For example what is an element $b\in B$? Or how should I think of $B$?

Any insight would be useful - so if a nice description exist when $R$ is say, perfect, or a complete DVR, or even a field, you can assume that.

My main interest is for general $p$-divisible groups, however if some nice insight can be given for $p$-divisible groups arising from elliptic curves or abelian varieties I'd appreciate it.

I have some grasp for connected $G$, because in this case $G$ corresponds to a formal Lie group. So if, say, $G[p^n]=\text{Spec}(B_n)$, and if $A=R[[X_1,\ldots,X_n]]$ is the corresponding power series ring of the formal Lie group, then $B=A/J$ with $J$ being the ideal generated by $X_i^p$ in $A$, where $X_i^p$ is viewed as the result of the formal group law applied $p$ times to $X_i$ (this is detailed in Tate's "$p$-divisible groups" for $G[p^n]$, section (2.2); note that it is assumed there that $R$ is complete, noetherian, local with residue field of char $p$).

But for more general $G$ (e.g. which doesn't split into a connected and etale component, for example) I don't know what happens.

Thank you.

p.s. why is there no "$p$-divisible-groups" tag? :)

• there's already a "formal-group" tag, which is already quite focussed (49 questions as of now since creation of MO); no need to create an even more specific tag.
– YCor
Apr 26, 2017 at 11:24