[Math] Does closed immersion imply projective morphism

Question

Here $X\to Y$ is a projective morphism means: $X\to Y$ factors through a closed immersion $X\to \mathbb{P}_{Y}^{m}$, and then followed by the projection $\mathbb{P}^{m}_{Y}\to Y$. I have no idea how to find this $\mathbb{P}_{Y}^{m}$.

Answer 1

A finite morphism is proper. So we know that a closed immersion is finite and therefore is proper.

Alternatively, just take $m = 0$ and then $$\mathbf{P}^0 = \textrm{Proj } \mathbf{Z} = \textrm{Spec } \mathbf{Z}$$