Assume $R$ is a Henselian local ring with $k=R/m_R$ ($m_R$ is the unique maximal ideal of $R$) and $G$ a finite flat group scheme over $R$. We denote by $G_k= G \otimes_R k$ the closed fiber.

There is a well known result that states that there exist an exact sequence of finite flat group schemes over $R$: $$\tag{$*$} 1 \to G^0 \to G \to G^{\text{ét}} \to 1$$ where $G^0$ is the neutral component of $G$ (= component of "neutral element" $e \in G$) and $G^{\text{ét}}$ is finite étale over $R$.

For such $G$ Grothendieck's version of Hensel's Lemma holds (EGA IV 4: 18.5.19): $$\{\text{connected components of $G$}\}= \{\text{connected components of $G_k$}\}.$$ I'm familar with some "conventional" ways to prove the existence of the sequence $1 \to G^0 \to G \to G^{\text{ét}} \to 1$ which all make use of "classical" formulation of Hensel's lemma (the lifting roots of polynomials $\bar{f} \in k[x]$ to those of $f \in R[x]$).

I'm really curious if there exist a proof that is more "geometric" proof in the sense that it uses Grothendieck's version of Hensel's Lemma to show the existence & exactness of sequence above.


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