TITLE: Is a proper morphism between projective schemes projective morphism？
QUESTION [6 upvotes]: That is if $f :P^n_ {X} \rightarrow P^m_{Y}$ is a proper morphism, then it is a projective morphism.
Projective morphism means that it factors through projective scheme of last term and the first component is a closed immersion.

REPLY [7 votes]: Let $f:X\to Y$ be a (not necessarily proper) morphism of projective schemes over a field $k$ (the same works over a scheme $S$). Then $f$ is projective.
Proof: $X\subseteq \mathbb{P}^n_k$ (closed immersion) for some $n$. Then consider the closed immersion $X\subseteq X\times_k Y$ in the graph, which is a closed subscheme of $\mathbb{P}^n_k\times_kY=\mathbb{P}^n_Y$ (here I am using that projective morphisms are proper, which is a standard application of the valuative criterion for properness). This gives the required factorization $X\to \mathbb{P}^n_Y\to Y$.