Recently I am studying the display theory of formal p-div groups ([1] )by Zink.
I would like to study by working on an example. As far as I understood, the display theory is a generalization of Dieudonne Module. Zink said in ([1] ) Page 6 an exact sequence
$$ 0\rightarrow \mathbf{G}_{\mathcal{P}}^{-1}\rightarrow\mathbf{G}_{\mathcal{P}}^{0}\rightarrow \mathrm{BT}_{\mathcal P}\rightarrow 0 $$
to associate a 3n-display $\mathcal P=(P,Q,F,V)$ to a p-divisible group $\mathrm{BT}_{\mathcal P}$. Roughly speaking, the functor $\mathbf{G}_{\mathcal{P}}^{-1}$ and $\mathbf{G}_{\mathcal{P}}^{0}$ is defined on the category $\mathrm{Nil}_R$ of Nilpotent $R$-algebra and for every object $\mathcal N\in \mathrm{Nil}_R$
$$ \mathbf{G}_{\mathcal{P}}^{0}\left(\mathcal N\right)= \widehat{W}(\mathcal N)\otimes_{W(R)}P $$
and $\mathbf{G}_{\mathcal{P}}^{-1}$ is some subfunctor of $\mathbf{G}^0_{\mathcal P}$(on the top of page 6 of [1] ).
My Question is , how this construction works for the multiplicative formal group $\widehat{\mathbb{G}}_m$ over $\overline{\mathbb F}_p$. The display for $\widehat{\mathbb{G}}_m$ should be $\mathcal P=(P,Q,F,V)$ with $P=Q=W(\overline{\mathbb F}_p)$, $F=p$ and $V=1$. How can we describe the following map?
$$ \mathbf{G}_{\mathcal{P}}^{0}\left(\mathcal N\right)= \widehat{W}(\mathcal N)\otimes_{W(R)}P\rightarrow \widehat{\mathbb{G}}_m(\mathcal N) $$
By the way, can we reconstruct the formal group law of $\mathbb{G}_m$ by the display? Let us say $R=\overline{\mathbb{F}_p}$ and $\mathcal N= pW(\overline{\mathbb{F}_p})/p^3$.
