Motivation of separated scheme

Question

A scheme is called separated if it is separated over $\mathbb{Z}$. Is there a specific reason to define like this, i.e. separated over $\mathbb{Z}$? I know Spec$\mathbb{Z}$ is a final object. But are there any other reasons or motivations?

Answer 1

Here's a nice fact which might be convincing:

If a scheme $X$ is separated over $\operatorname{Spec}\Bbb Z$ then for every scheme $Y$ and every morphism $f:X\to Y$, the morphism $f$ is separated.

Here's one way to prove this: there is a general fact that if $f:X\to Y$ and $g:Y\to Z$ are morphisms of schemes, and $P$ is a property of morphisms preserved under composition and base change, then under the hypothesis that $g\circ f$ has property $P$ and the diagonal of $g$ has property $P$, we can conclude $f$ has property $P$. In our case, we can take $P$ to be the property of separatedness, and then for $Z=\operatorname{Spec}\Bbb Z$, $g\circ f$ must be the unique morphism $X\to\operatorname{Spec}\Bbb Z$ (which is separated by assumption), and the diagonal of $g$ is separated because it is a locally closed immersion (this is true of the diagonal of any morphism), so we conclude $f$ is separated.

Answer 2

One motivation can be given from topology: A space $X$ is called hausdorff, if for every pair of points $x \neq y$, there are open sets $x \in U$ and $y \in V$ such that $U \cap V = \emptyset$. Then it is a good exercise to show that $X$ is hausdorff if and only if the diagonal map $\Delta: X \to X \times X$ is closed.

Actually, the analogy goes a bit further: A classical example of a non-hausdorff space is the real line with double $0$, that is obtained by gluing two copies of $\mathbb{R}$ along the set $\mathbb{R} \setminus \{0\}$. The algebro-geometric analogue is the variety obtained by gluing two copies of $\mathbb{A}^n$ along $\mathbb{A}^n \setminus \{0\}$.