In Hartshorne (and pretty much everywhere else I know) a proper morphism is defined to be
A morphism of schemes $f:X\to Y$ is said to be proper if it is sperated, of finite type and universally closed.
But then I started reading "Introduction to Moduli Problems and Orbit Spaces" by Newstead. He defines a proper morphism between two separated varieties (which he simply calls variety, calling non-seperated varieties prevarieties) quite differently:
Let $X,Y$ be (separated) varieties. A morphism $f:X\to Y$ is proper if for any other (separated) variety $Z$, the map $f\times 1: X\times Z\to Y\times Z$ is closed.
I can't figure out why these two definitions are equivalent to each other when the Harthsorne definition is restricted separated varieties over $k$. The finite type in the definition of Hartshorne is clearly redundant when dealing with varieties.
Edit: After a few comments, I think I should say what parts of this equivalence is problematic for me to prove (although the comments already helped me resolve parts of the problem/confusion).
- If $f:X\to Y$ is universally closed, i.e. for all separated $k$-varieties $Z$, the projection map $Z\times_Y X\to Z$ is closed (after base change), then $f\times 1: X\times Z\to Y\times Z$ is closed:
Consider the base change of $f:X\to Y$ under the projection map $\pi_Y: Z\times Y\to Y$ with $Z$ arbitrary. One easily check that $X\times Z$ with $\pi_X: X\times Z\to X$ and $f\times 1: X\times Z\to Y\times Z$ satisfies the UMP of fibered product. Then universally closed yields $f\times 1$ is closed.
So what remains to prove is
If $f\times 1: X\times Z\to Y\times Z$ is closed for all separated $k$-varieties $Z$, then $Z\times_Y X\to Z$ is closed for all $Z$ too.
Also I can really get some help on why any morphism of separated $k$-varieties is separated. I think it should be simple and I'm just thinking over-complicated.