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There are two definitions of a projective morphism.

Hartshorne: A morphism $f: X\to Y$ is projective if it factors as $f=gi$, where $i: X\to P_Y^n$ is closed imbedding and $g: P_Y^n\to Y$ is canonical.

Ravi's notes: Morphisms of the form $Proj(L)\to Y$ where $L$ is a quasicoherent sheaf of graded $O_Y$-algebras, finitely generated in degree 1.

In Ravi's notes, it is shown that finite morphisms are projective. The corresponding algebraic fact is that $Proj(A\oplus B\oplus B\oplus B\cdots)=Spec(B)$ for a finite morphism $A\to B$ of rings. In Ravi's notes, it is also mentioned that a finite morphism need not be projective in Hartshorne's sense. Is there an easy counterexample showing this?

I apologize if this turns out to be a simple question.

best, minimax

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This is not a simple question ! – user18119 Dec 10 '12 at 23:14

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