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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. |
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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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