Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
A quasi-projective morphism of locally Noetherian schemes is of finite type, so that should be a hypothesis as well. –  Matt Sep 22 '11 at 0:59
    
Oh, you are right! I will edit the condition. Thank you so much. –  Hiro Sep 22 '11 at 1:12
add comment

1 Answer

up vote 6 down vote accepted

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.

share|improve this answer
    
Thanks for your proof and the reference! –  Hiro Sep 22 '11 at 1:50
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.