An example of $p$-divisible group from the book of Demazure. I have some questions when I read the book "Lectures on $p$-divisible groups". Precisely, my questions are in (page 76) chapter IV, section 3: The $F$-spaces $E^{\lambda}, \lambda\geq 0$.
the problems in summary contain computations of kernel of maps of group schemes, invariants related with $p$-divisible groups: Dieudonné modules, Hopf algebras, height, dimension...
I put the questions as follows (should be classical ones as it is just quickly mentioned in the book) with some of my attempts (please correct the mistakes I possibly made):
Question 1. In page 76, $\overline{M}^{\lambda}:=\mathbb{Z}_p[F]/(F^{r-s}-V^s)$ is defined, which I am already confused with:
1.1 Maybe $\overline{M}^{\lambda}:=\mathbb{Z}_p[F, V]/(F^{r-s}-V^s, FV=p, VF=p)$? But this module have elements of the form, say $V^n$ with $2\leq n\leq s-1$, which not seems not expected in the original expression.
1.2 Or maybe for some reasons we can just put $V:=pF^{-1}$ and somehow it makes sense in $\mathbb{Z}_p[F]$?
1.3 Remark that $\mathbb{Q}_p\otimes \overline{M}^{\lambda}$ makes "more sense" for me. I can look at $\mathbb{Q}_p\otimes M^{\lambda}=E^{\lambda}=\mathbb{Q}_p[T]/(T^r-p^s)$: as $p^{-1}$ exists, hence $T^{-1}$ exists. Hence change $T$ by $F$ we can rewrite the module as
\begin{align*}
\mathbb{Q}_p[F]/(F^r-p^s)&=\mathbb{Q}_p[F]/(F^{-s}(F^r-p^s))\\
&=\mathbb{Q}_p[F]/(F^{r-s}-(pF^{-1})^s)\\
&=\mathbb{Q}_p[F]/(F^{r-s}-V^s)
\end{align*}
where $V:=pF^{-1}$.
This computation some how "convince" me with the statement in the book that "$\overline{M}^{\lambda}$ is a lattice in $E^{\lambda}$", but still question 1 have to be answered.
Question 2 (main question). In the same paragraph as question 1, it is claimed that $\overline{M}^{\lambda}$ is a Dieudonné module, precisely it is the associated Dieudonné module of the following constructed $p$-divisible group:
Let $G^{\lambda}$ be the $p$-divisible group over $\mathbb{F}_p$ defined by the exact sequence
$$ 0\to G^{\lambda} \to W(p) \xrightarrow{F^r-V^s} W(p)  $$
where $W(p)=\varinjlim(\mathbf{Ker}p^n: W_{\mathbb{F}_p}\to W_{\mathbb{F}_p})$. In other words, the statement is: $M(G^{\lambda})=\overline{M}^{\lambda}$ (the Dieudonné module) and $E(G^{\lambda})=E^{\lambda}$ (the $F$-space).
2.1 How to look at $W(p)=\varinjlim(\mathbf{Ker}p^n: W_{\mathbb{F}_p}\to W_{\mathbb{F}_p})$? Is it a $p$-divisible group? 
(seems not: as I computed below, it looks not like a direct limit of finite group schemes, at least.)
2.1.1 How should I describe the group scheme $\mathbf{Ker}p^n: W_{\mathbb{F}_p}\to W_{\mathbb{F}_p}$? Can I describe in the following way:
For $W_{\mathbb{F}_p}$, as group scheme I may write $W_{\mathbb{F}_p}=\prod_{n\geq 0}\mathbb{A}^1_{\mathbb{F}_p}=\mathbf{Spec}  \mathbb{F}_p[X_0, X_1, \cdots]$. Remark that we are working in characteristic $p$ case and hence we have $p^n: W_{\mathbb{F}_p}(R)\to W_{\mathbb{F}_p}(R)$ for some $\mathbb{F}_p$-algebra $R$ is of the form $(x_0, x_1, \cdots)\mapsto (0, \cdots, 0, x_0^{p^n}, x_1^{p^n}, \cdots)$ where $x_0^{p^n}$ is in the $n+1$-th position. (From this, by mimic the situation in algebraic geometry, the morphism on the coordinate ring should be $f: \mathbb{F}_p[X_0, X_1, \cdots]\to \mathbb{F}_p[Y_0, Y_1, \cdots]: f(X_0)=\cdots =f(X_{n-1})=0, f(X_n)=Y_0^{p^n}$, $f(X_{n+1})=Y_1^{p^n}\cdots$) hence the kernel of $p^n$ described by should be
$\require{AMScd}$
\begin{CD}
\mathbf{Ker}p^n @>{}>> W_{\mathbb{F}_p}\\
@VVV @VVp^nV\\
\mathbf{Spec}\mathbb{F}_p @>{}>> W_{\mathbb{F}_p}
\end{CD}
which corresponds to
$\require{AMScd}$
\begin{CD}
 \mathbb{F}_p[X_0, X_1, \cdots]\otimes \frac{\mathbb{F}_p[X_0, X_1, \cdots]}{I}  @<{}<<  \mathbb{F}_p[X_0, X_1, \cdots] \\
@AAA @AAp^nA\\
\mathbb{F}_p @<{}<< \mathbb{F}_p[X_0, X_1, \cdots]
\end{CD}
with $I=(X_0^{p^n}, \cdots, X_m^{p^n}, \cdots)$. Hence the kernel of $p^n$ is like $W_n(p)=\mathbf{Spec}\frac{\mathbb{F}_p[X_0, X_1, \cdots]}{(X_0^{p^n}, \cdots, X_m^{p^n}, \cdots)} $ and the limit is finally again $\mathbf{Spec}\mathbb{F}_p[X_0, X_1, \cdots]$ or maybe $\mathbf{Spec}\mathbb{F}_p[\![X_0, X_1, \cdots]\!]$ or else?
2.1.1.1 As this is not finite group, what is $\varinjlim \mathbf{Spec}\frac{\mathbb{F}_p[X_0, X_1, \cdots]}{(X_0^{p^n}, \cdots, X_m^{p^n}, \cdots)}$?
(Is it $\mathbf{Spec} \varprojlim\frac{\mathbb{F}_p[X_0, X_1, \cdots]}{(X_0^{p^n}, \cdots, X_m^{p^n}, \cdots)}$? This may not be as in the formal group case, where the Hopf algebra is the inverse limite of the Hopf algebra of the finite group schemes. What is $\varprojlim\frac{\mathbb{F}_p[X_0, X_1, \cdots]}{(X_0^{p^n}, \cdots, X_m^{p^n}, \cdots)}$? Can I relate $\mathbf{Spec} (1+\mathbb{F}_p[\![t]\!])$ with $W_{\mathbb{F}_p}$ using Artin-Hasse exponential?)
2.2
How could I compute the Dieudonné module of $G^{\lambda}$?
2.2.1
How could I compute the Dieudonné module of $W(p)$? Since I may try to compute it by the exact sequence
$$ M(W(p))\xrightarrow{V^r-F^s} M(W(p))\to M(G^{\lambda})\to 0$$
2.2.2 How to calculate $M_n(W(p))$? As $M_n(W(p))\simeq M(W_n(p))$, hence how to
compute $M(W_n(p))$?
(Having $W_n(p)=\mathbf{Spec}\frac{\mathbb{F}_p[X_0, X_1, \cdots]}{(X_0^{p^n}, \cdots, X_m^{p^n}, \cdots)}$, it is not clear how can I have the claimed Dieudonné module.)
2.3 Finally, with the correct Dieudonné module we can see the correct height (which is the rank of the Dieudonné module) and correct dimension (which is the dimension of as formal group). How to see the dimension is $s$ as claimed?
Thanks in advance for any remarks on any of these questions; the problems are in summary: the computations of kernel of maps of group schemes, invariants related with $p$-divisible groups --- some nontrivial examples (even hard to find), Dieudonné modules, Hopf algebras... Any general remarks or good references are also welcome.
 A: I summarize a partial answer here (with the help of many people)
Notations:
$$D_k=W(k)[F, V]/(Fa=a^pF, aV=Va^p, FV=VF=p \text{ for all } a\in W(k) )$$
$$W_k=\mathsf{Spec}(k[t_0, t_1, \cdots])$$
$$K_n:=\mathbf{Ker}(p^n\cdot id_{W_k})\subset W_k$$
$$W(p)_{k}=\varinjlim_n K_n$$
$${}_{n}W_m=\mathsf{Spec}(k[t_0, \cdots, t_{n-1}]/\big( t_0^{p^m}, \cdots, t_{n-1}^{p^m}  \big))$$
Lemma:
Let $k$ be a perfect field and $f(x, y)\in W(k)[x,y]$ a polynomial with $deg_{y}f>0$. The formal group $G=\mathsf{Ker}(f(F, V):W(p)_k\to W(p)_k)$ is a $p$-divisible group with Dieudonné module
$$ M(G)=\varprojlim_n D_{k}/(F^n, f(F, V)).$$
Proof:
Put $G_n=\mathsf{Ker}(f(F, V):K_n\to K_n)$, then $G=\varinjlim_{n} 
 G_n$
(as $\{ K_n \}_{n\geq 0}$ form a filtered system, its injective limit preserves kernel).
We define $G_{n,m}$ by the following exact sequence
$$0\to G_{n,m} \to {}_{n}W_m \xrightarrow{f(F, V)} {}_{n}W_m. $$
Taking the inverse limit (remark that inverse limit commutes with kernel) and we have $G_{n}=\varprojlim G_{n,m}$.
We know that $M({}_{n}W_m)\simeq D_{k}(F^n, V^m)$, and hence we have
$$ M({}_{n}W_m)\xrightarrow{f(F, V)} M({}_{n}W_m) \to M(G_{n, m}) \to 0, $$
and hence $M(G_{n,m})= D_k/(F^n, V^m, f(F, V))$.
Then we have
$$M(G_n)=\varinjlim_m M(G_{n,m})=\varinjlim_m D_k/(F^n, V^m, f(F, V))$$
$$= D_k/(F^n, f(F, V)).$$
Hence
$$ M(G)=\varprojlim_n D_{k}/(F^n, f(F, V)).$$
This gives the Dieudonné module of $G$.
Conversely, by the equivalence of category given by Demazure (in page 71), we can construct a $p$-divisible group $G^{'}:=\varinjlim_n G_n^{'}$ where $G_n^{'}$ is defined to be the group scheme whose Dieudonné module is $M(G)/p^n$. Then we check that $G=G^{'}$.
(Here needs some details: We can see that $M(G_{n(r+s)}^{'})\twoheadrightarrow M(G_{n(r+s)}) \twoheadrightarrow M(G_n^{'})$ and hence $M(G)=M(G^{'})$, but then how to conclude? Shouldn't we have even $G^{'}_n=G_n$, as both should be $G[p^n]$?)
Example:
When $k=\mathbf{F}_p$, and $f(F, V)=F^r-V^s$, we have
$$ M(G)=\mathbf{Z}_p[F, V]/(FV-p, VF-p, F^r-V^s). $$
