An étale morphism that restricts to an isomorphism on a closed subvariety.

Question

If an étale morphism $f:X\rightarrow Y$ induces an isomorphism $f:f^{-1}(Z)\rightarrow Z$ for some closed subvariety $Z$ of $Y$. Doesn't it imply that $f$ is an isomorphism (I believe the answer should be negative because it is assumed to be so in a paper, but cannot come up with an example.)

Answer 1

This condition does not imply that $f$ must be an isomorphism. Let $k$ be a field of characteristic not two, and consider the following composition: $$\Bbb A^1_k\setminus \{0,-1\} \hookrightarrow \Bbb A^1_k\setminus \{0\} \stackrel{\cdot^2}{\to} \Bbb A^1_k\setminus \{0\}$$ The first map is an open immersion, so it is etale. The second map is the squaring map, which is etale. Taking $Z=\{x=1\}$ in $\Bbb A^1_k\setminus \{0\}$, we have that $f:f^{-1}(Z)\to Z$ is an isomorphism, but $f$ is not an isomorphism.