Let $R$ be a commutative ring, $p >0$ prime and $G$ a finite, locally free group scheme over $R$ of rank $p^n$; $n \in \mathbb{N}\_{\ge 1}$. Assume $p \in R^\*$ (i.e. is a unit in $R$). 


Question: Why this condition on the rank implies that $G$ is étale?


By definition etale is equivalent to flat & unramified. As $G$ is locally free it's obviously flat. Be unramified is also a local condition. Thus we can translate the problem to commutative algebra and asking why the free $R$-module $R^{p^n}$ is unramified at a prime $\mathfrak{q} \subset R$ if $p \in R^\*$.



Locally free group scheme étale
This is not so easy, but relies on a well-known structure theorem for connected group schemes over a perfect field.


**Lemma 1.** *A finitely presented morphism $Y \to X$ of schemes is unramified if and only if $Y\_x \to x$ is unramified for all $x \in X$.*


*Proof.* See [EGA IV$\_4$, Cor. 17.4.2]. $\square$


Thus, we may reduce to the case $R = k$ for $k$ a field, and by flat descent to the case where $k$ is algebraically closed. Then unramified for a finite extension means (geometrically) reduced.


**Lemma 2.** *A finite group scheme over an algebraically closed field is a semidirect product of an étale group scheme and a connected group scheme.*


*Proof.* See for example [Wat, §6.8]. The étale part is $\pi\_0(G)$ and the connected part is $G^0$. $\square$


**Theorem.** *If $G$ is a geometrically connected finite group scheme over a perfect field $k$ of characteristic $p > 0$, then $\Gamma(G,\mathcal O\_G)$ is isomorphic to $k[X\_1,\ldots,X\_n]/(X\_1^{p^{e\_1}},\ldots,X\_n^{p^{e\_n}})$ for some $e\_1,\ldots, e\_n \in \mathbf Z\_{>0}$.*


*Proof.* See for example [Wat, §14.4]. I don't know a quick summary of why this is supposed to be true (I would be interested if someone does), but the proof is not that hard. $\square$


**Theorem.** *If $G$ is group scheme over a field of characteristic $0$, then $G$ is geometrically reduced.*


*Proof.* See for example [Wat, §11.4]. $\square$


In particular, if the rank of $G$ not divisible by $p$ (e.g. $p = 0$), then $G^0$ has to be trivial and $G$ is étale. $\square$




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**References.**


[EGA IV$\_4$] A. Grothendieck, [*Éléments de géométrie algébrique. IV: Étude locale des schémas et des morphismes de schémas (Quatrième partie).*](http://www.numdam.org/item?id=PMIHES_1967__32__5_0). Publ. Math., Inst. Hautes Étud. Sci. **32**, p. 1-361 (1967). [ZBL0153.22301](https://zbmath.org/?q=an:0153.22301).


[Wat] W.C. Waterhouse, [*Introduction to affine group schemes*](https://doi.org/10.1007/978-1-4612-6217-6). Graduate Texts in Mathematics **66** (1979). [ZBL0442.14017](https://zbmath.org/?q=an:0442.14017).


