# Compactifiable morphisms

Let's say a morphism $f:X\to Y$ is compactifiable if it admits a factorization $f = pj$ with $j:X\to P$ an open immersion and $p:P\to Y$ proper.

In SGA 4 Exp. XVII, Deligne says that Nagata proved that any morphism of separated integral northerian schemes is compactifiable but that he didn't understand the proof.

My questions:

• Where can I find a proof of Nagata's theorem?
• What about the complex analytic setting?
• I just noticed that your question was "Where can I find a proof that any algebraic morphism of quasi-projective varieties is compactifiable?". This is pretty much obvious; just embed $X$ into $\mathbb P^N$, and take the closure of $X$ in $\mathbb P^N \times Y$. – Angelo Apr 24 '10 at 13:55
• Obvious indeed. Changed the question. – AFK Apr 24 '10 at 14:13
• Nagata's theorem doesn't need the "integral" hypothesis (even though Nagata's original formulation may have had it, possibly due to his use of earlier style of alg. geometry; I don't remember). The references mentioned below avoid it. – BCnrd Apr 24 '10 at 16:34