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Let $\phi:X\to Y$ be an étale morphism of Noetherian schemes. Does $\phi$ have to be quasi-affine? In other words, if $Y$ is affine does it mean that $X$ is quasi-affine?

It will follow from the fact that quasi-finite morphisms are quasi-affine, but I do not know whether this is true.

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    $\begingroup$ $\phi$ will be quasi-affine iff it is separated, which is not automatic for etale morphisms. The nontrivial implication is Zariski's Main Theorem. $\endgroup$
    – Marc Hoyois
    Commented Jun 6, 2018 at 0:45
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    $\begingroup$ Silly example: the line with two origins projecting to $\mathbb{A}^1$ is étale (it's even a local isomorphism!) but it is certainly not quasi-affine $\endgroup$
    – Denis Nardin
    Commented Jun 6, 2018 at 11:32
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    $\begingroup$ As Marc Hoyois indicates: a quasi-finite morphism is quasi-affine if and only if it separated. To prove this, you can use ZMT (as he says). Indeed, note that a quasi-finite separated morphism factors as an open immersion and a finite morphism. Since finite morphisms are affine, it follows that every quasi-finite separated morphism is quasi-affine. $\endgroup$
    – Ariyan Javanpeykar
    Commented Jun 6, 2018 at 12:40

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