I am reading (parts of) the paper "Moduli of $p$-divisible groups" by Scholze and Weinstein, and I am stuck at understanding the definition of level structures in Rapoport-Zink spaces (cf. Definition 6.3.1 in the paper).
Let me briefly explain the context:
Let $k$ be a perfect field of characteristic $p>0$ and fix a $p$-divisible group $H$ of dimension $d$ and height $h$ over $k$. The functor $\mathcal{M}$ on the category $\operatorname{Nilp}_{W(k)}$ of $W(k)$-algebras on which $p$ is nilpotent given by
$$
R \mapsto \text{Isomorphism classes of deformations } (G,\rho) \text{ of }H \text{ to } R
$$
is representable by a formal scheme $\mathcal{M}$ over $\operatorname{Spf}W(k)$ locally admitting a finitely generated ideal of definition. Here a pair $(G,\rho)$ consists of a $p$-divisible group $G$ over $R$ and a quasi-isogeny $\rho\colon H\otimes_k R/p \to G\otimes_R R/p$.
Passing to the adic generic fibre we obtain the functor $\mathcal{M}_{\eta}^{\operatorname{ad}}$ on the category of complete Huber pairs over $(W(k)[p^{-1}],W(k))$, which is given as the sheaf associated to the presheaf $$ (R,R^+)\mapsto \varinjlim_{R_0\subset R^+} \mathcal{M}(R_0), $$ where the colimit runs over the open and bounded $W(k)$-subalgebras of $R^+$. Here, $\mathcal{M}(R_0)=\varprojlim_n \mathcal{M}(R_0/p^n)$ coincides with the set of isomorphism classes of deformations of $H$ to $R_0$.
Now the paper defines a level-$n$ structure on a section $(G,\rho)\in \mathcal{M}_{\eta}^{\operatorname{ad}}(R,R^+)$ to be a morphism of $\mathbb{Z}/p^n$-modules $$ \alpha\colon (\mathbb{Z}/p^n)^h\to G[p^n]^{\operatorname{ad}}_{\eta}(R,R^+) $$ which becomes an isomorphism at every point of $\operatorname{Spa}(R,R^+)$.
$\textbf{Question}$: How is $G[p^n]^{\operatorname{ad}}_{\eta}(R,R^+)$ defined? The notation suggests that $G[p^n]_{\eta}^{\operatorname{ad}}$ is in fact a sheaf on the category of complete Huber pairs over $(W(k)[p^{-1}],W(k))$, so I might as well ask: how is $G[p^n]_{\eta}^{\operatorname{ad}}$ defined?
The section $(G,\rho)$ is given by an open covering $\cup_i\operatorname{Spa}(R_i,R_i^+)=\operatorname{Spa}(R,R^+)$, open and bounded $W(k)$-subalgebras $R_{i,0}\subset R_i^+$ and deformations $(G_i,\rho_i)\in \mathcal{M}(R_{i,0})$, compatible on overlaps.
$\textbf{Subquestion}$: Each $R_{i,0}$ is $p$-adically complete and $G_i[p^n]$ is representable by a finite locally free $R_{i,0}$-group scheme; however, it seems to me that one wants to take into account the topology on $R_{i,0}$, and so $G_i[p^n]$ should be something formal affine (compare the proof of Proposition 3.3.2). So my guess is that one $\textit{defines}$ $G_i[p^n]:=\varinjlim_m (G_i\otimes R_{i,0}/p^m)[p^n]$, which is equal to the formal spectrum of the ring representing $G_i[p^n]$ (not taking into account the topology) endowed with the $p$-adic topology. Is this correct?
From the formal scheme $G_i[p^n]$ one can pass to the adic generic fibre $G_i[p^n]_{\eta}^{\operatorname{ad}}$, and I guess that $G[p^n]_{\eta}^{\operatorname{ad}}$ is somehow obtained from the $G_i[p^n]_{\eta}^{\operatorname{ad}}$, but I don't know how.
A clear definition of the indicated object of interest is highly appreciated.
