Let $R$ be a commutative complete local ring with residue field of
characteristic $p > 0$.
A $p$-divisible group over $R$ of height $h$ is an inductive system
$(G_{\nu}, i_{\nu})$
for which following properties hold:
- (1) $ G_{\nu}$ is a finite, flat, commutative group scheme over $R$ of order $p^{h \nu}$.
- (2) For each $ \nu$, we have the exact sequence
$$ 0 \to G_{\nu} \xrightarrow{i_{\nu}} G_{\nu+1} \xrightarrow{p^{\nu}} G_{\nu+1} $$
that is, $i_{\nu}$ maps $G_{\nu}$ (as closed immersion) to
$ G_{\nu+1}$ and identifies it with the
kernel of $ p^{\nu}$ on the latter group scheme.
The inductive system $(G_{\nu}, i_{\nu})$ gives a formal group;
Note that by finiteness over $R$ assumption all members $ G_{\nu} = \operatorname{Spec} A_{\nu} $ are affine and we
denote by $A = \varprojlim_{\nu} A_{\nu} $,
the formal spectrum $G= \operatorname{Spf} A$ associated to the
$p$-divisible group $(G_{\nu}, i_{\nu})$.
We assume in the following that all members $G_{\nu}$ to be affine.
In Cornell's & Silverman's Arithmetic Geometry, p. 65 first line, it is stated without proof that
if $G$ is connected (ie every member $ G_{\nu} = \operatorname{Spec} A_{\nu} $ is connected as $R$-scheme), then $A \varprojlim_{\nu} A_{\nu} $ is
isomorphic to the formal power series ring $R[[X_1,..., X_n ]]$.
Any ideas on how to verify this claim?