(a) Let $f:X \rightarrow S$ be a separated morphism of schemes. Show that for any subscheme $U \subset X$, the restriction $f\mid_{U}:U \rightarrow S$ is separated.
(b) Let $R$ be a commutative ring and $X$ be an $R$-scheme. Show that $X$ is separated over $R$ i.e. the structure map $X \rightarrow \textrm{Spec} R$ is separated iff $X$ is separated over $\mathbb{Z}$.
(c) Let $f:X \rightarrow Y$ be a separated morphism of schemes. Show that for any $y \in Y$, the fiber $X_{y}$ is separated over $k(y)$, where $k(y)$ is the residue field at $y$.
I recently encountered these exercises in my schemes reading. I am wondering how I might approach these. Any help would be appreciated!
Best Answer
In general, separated morphisms are stable under composition and under base change. This follows from the definitions, no valuative criterion is necessary here. You can also find this in every detailed introduction to schemes. This implies (c). Open immersions are seperated (in fact, every monomorphism is separated). This implies (a). Affine morphisms are separated. This gives the one direction of (b). Now assume that $X$ is separated over $R$, i.e. $X \to X \times_R X$ is a closed immersion. Then for every pair of open affines $U,V \subseteq X$ we have that $U \times_R V$ is affine, hence the preimage $U \cap V$ is affine and $\Gamma(U \times_R V) = \Gamma(U) \otimes_R \Gamma(V) \to \Gamma(U \cap V)$ is surjective. But this implies that also $\Gamma(U) \otimes_\mathbb{Z} \Gamma(V) \to \Gamma(U \cap V)$ is surjective. From this we see that $X$ is separated over $\mathbb{Z}$. Actually a more general statement is true: If $f \circ g$ is separated, then $g$ is separated (the standard proof factors $g$ over its graph).