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Consider the exact sequence of finite flat group schemes over the $2$-adic integers ring $\mathbb{Z}_2$: $$0\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow A\longrightarrow\mathbb{Z}/2\mathbb{Z}\longrightarrow0.$$

$\textbf{Question}$: Why is $A$ étale?

The motivation is that if I know $A$ is étale, but we also know $\mathbb{Z}_2$ is a complete DVR, and any finite étale group scheme over such a ring is constant, so I can derive $A$ is a constant group scheme.

Thanks!

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    $\begingroup$ The map $A \to \mathbf{Z}/2$ is a finite \'etale map (as it is the quotient by a finite \'etale subgroup scheme), and the target is \'etale over $\mathbf{Z}_2$, so $A$ is also \'etale over $\mathbf{Z}_2$. $\endgroup$ Oct 4, 2019 at 15:14
  • $\begingroup$ "... $\mathbb{Z}_{2}$ is a complete DVR, and any finite étale group scheme over such a ring is constant" Do you need more conditions on the DVR? What about $\mu_{5}$ over $\mathbb{Z}_{2}$? $\endgroup$ Oct 5, 2019 at 10:59
  • $\begingroup$ @user2831784 I remember that the residue field of the DVR should be algebriac closed. But I have seen somewhere such an exact sequence makes $A$ a constant scheme. Sorry, I don't know much about these, I am trying to understand Mazur's theorem about elliptic curves. $\endgroup$
    – user141691
    Oct 5, 2019 at 14:09

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