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I'm reading Mumford's Red Book and I'm having issues with his definition of complete variety (Chap.1.9),

In the introductory paragraph he talks about impossibility of birational embedding in larger varieties for projective varieties. He defines birational morphisms of varieties as dominant morphisms that induce an isomorphism between the function fields of the varieties.

Then he gives the definition of completeness for a variety, that is that $\forall Y$ variety, $p_2:X \times Y \rightarrow Y$ is a closed map. Is the property of completeness enough for not having birational inclusions? I tried using the closed diagonal property and the theorem of Chap 1.8 but I got nowhere. It seems strange to me. Thank you in advance!

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    Using this property, can you show any morphism $f:X\to Z$ is a closed map? In particular, if this was a birational inclusion, then the image is closed and thus $f$ is an isomorphism. – 2017-02-06
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    I can show that any map is closed, combinig the diagonal injection and the projection.But I don't believe that makes f an isomorphism. It seems to me just a bijective birational morphism. Can you show me that the inverse is a morphism between varieties? i.e. $ (f^{-1})^*$:$\Gamma(X, o_X) \rightarrow \Gamma(Y, o_Y)$ is a morphism? – 2017-02-06
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    Any surjective open immersion is an isomorphism. – 2017-02-06
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    Can you link the chapter or the proof? Thank you – 2017-02-07
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    This is an elementary exercise. Why don't you try to prove it yourself? – 2017-02-07

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