What kind of conditions we need to make morphisms of schemes quasi-projective?

I am really interested in the following case:

If $f : X \to Y$ is an etale, of finite type and separated morphism of schemes, then is it quasi-projective?

If so, which conditions we use?

If necessary, please assume that the scheme $Y$ is locally noetherian.

  • $\begingroup$ A quasi-projective morphism of locally Noetherian schemes is of finite type, so that should be a hypothesis as well. $\endgroup$ – Matt Sep 22 '11 at 0:59
  • $\begingroup$ Oh, you are right! I will edit the condition. Thank you so much. $\endgroup$ – Hiro Sep 22 '11 at 1:12

The answer is yes, if you assume that $Y$ is quasi-compact, and $f:X\to Y$ is of finite type and separated.

Every etale morphism is unramified, which implies it is quasi-finite (Milne, Prop 3.2). This in turn implies by Zariski's main theorem (Milne, Thm 1.8), that $f$ factors as an open immersion followed by a finite map. Hence $f$ is quasi-affine, hence quasi-projective.

The reference is Milne: Etale cohomology.


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