# Level structures in deformation spaces of $p$-divisible groups

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.