Let $K$ be a $p$-adic local field with uniformizer $\pi \in \mathcal{O}_{K}$ and residue field $k = \mathcal{O}_{K}/\pi$. Let $G$ be a Lubin-Tate formal $\mathcal{O}_{K}$-module and $G_{0}$ its reduction to $k$.

Let $G_{0}[\pi]$ denote the $\pi$-torsion subgroup of $G_{0}$, this is a finite group scheme over $k$. The action of $\mathcal{O}_{K}$ on $G_{0}$ yields an action of the multiplicative monoid $\Gamma = (k, \cdot)$ on $G_{0}[\pi]$ and hence on its coordinate ring $\mathcal{O}_{G_{0}[\pi]}$.

**Question:** Does there always exist an element $g \in \mathcal{O}_{G_{0}[\pi]}$ which generates the coordinate ring as a module over the monoid ring $k[\Gamma]$? How to find such an element?

**Example:** Suppose that $K = \mathbb{Q}_{p}$, $\pi = p$ and $G = \mathbb{G}^{\wedge}_{m}$ the formal multiplicative group, so that

$\mathcal{O}_{G[p]} \simeq \mathbb{F}_{p}[x]/((1+x)^{p}-1)$.

Then an element $[\gamma] \in \Gamma = (\mathbb{Z}/p, \cdot)$ acts on $g = x+1$ via

$[\gamma]g = [\gamma](x)+[\gamma](1) = (x+1)^{\gamma}-1+1 = (x+1)^{\gamma}$

and hence the translates of $g$ along $0, \cdots, p-1 \in \mathbb{F}_{p}$ are of the form

$(x+1)^{0}, (x+1)^{1}, \cdots, (x+1)^{p-1}$

This is a $\mathbb{F}_{p}$-basis of $\mathcal{O}_{G_{0}[\pi]}$, so $g = x+1$ gives an element which generates the whole coordinate ring as needed.

**Some thoughts:** 1) In the case of $\mathbb{G}_{m}^{\wedge}$, where $\mathbb{G}_{m}[p] = \mu_{p}$, another way to find an element $g$ is to notice that Cartier duality gives an isomorphism

$\mathcal{O}_{\mu_{p}} \simeq \mathcal{O}_{\mathbb{Z}/p}^{*} \simeq Fun(\mathbb{Z}/p, \mathbb{F}_{p})^{*} \simeq \mathbb{F}_{p}[\mathbb{Z}/p]$,

(where $*$ denotes the dual vector space) so that the coordinate ring is just the regular representation of the multiplicative monoid $\Gamma = \mathbb{Z}/p$ and hence must be generated by a single element.

Also in the case of $\mathbb{G}_{m}^{\wedge}$, the invariant differential is given by $\omega(x) dx = \frac{1}{1+x} dx$. So perhaps it is possible to relate a generating element $g$ to invariant differentials?

We have verified that such an element also exists when $G_{0}$ is the Honda formal group law of height 2 over $\mathbb{F}_{4} \simeq W(\mathbb{F}_{4})/p$, and can be taken to be

$g = 1 + x + x^{2} + x^{3} \in \mathcal{O}_{G_{0}[p]} \simeq \mathbb{F}_{4}[x]/(x^{4})$