In Vakil's FOAG, definition 8.1.1 reads as
A morphism $\pi : X \to Y$ of schemes is a closed embedding if
- $\pi$ is affine, i.e., for every affine open subset $\mathrm{Spec} B$ of $Y$, $\pi^{-1}\mathrm{Spec} B \cong \mathrm{Spec} A$ is affine open in $X$.
- And the induced morphism $\mathrm{Spec} A \to \mathrm{Spec} B$ on global section is a surjective map $B \to A$
Then, in Exercise 8.1.A, I'm asked to show that the closed embedding $\pi$ identifies the topological space of $X$ with a closed subset of the topological space of $Y$.
The text uses the word "identify". I believe that by using "identify" he means "homeomorphic" as in the affine case: if we have a surjective ring map $B \to B/I$, then we have an induced map $\mathrm{Spec} B/I \to \mathrm{Spec} B$ which is a homeomorphism from $\mathrm{Spec} B/I$ to the closed subset $V(I)$ of $\mathrm{Spec} B$.
The question is I don't know how to show this.
I have some thoughts:
- I've managed to show that $\pi$ is injective: Take an affine open $\mathrm{Spec} B$ of $Y$, let $\mathrm{Spec} A = \pi^{-1}\mathrm{Spec}B$. By definition, on global sections, $\pi^\# : B \to A$ is surjective. Since we have the equivalence between category of affine schemes and rings, $\pi : \mathrm{Spec}A \to \mathrm{Spec}B$ is exactly induced by $\pi^\# : B \to A$. Since $\pi^\#$ is surjective, we know that $\pi^\#$ induces homeomorphism from $\mathrm{Spec}A$ to a closed subset of $\mathrm{Spec}B$, in particular, $\pi : \mathrm{Spec}A \to \mathrm{Spec}B$ is injective.
- By definition, $\pi$ is continuous, and we have just shown $\pi$ is injective. So it suffices to show that $\pi$ takes closed subsets of $X$ to closed subsets of $Y$.
By arguments above, we only know that $\pi$ takes each piece of some affine open cover of $X$ to some closed subset of affine open of $Y$. How can one take care of this gap, or should I try some different way?
Thank you for your help.
