I would like to prove that closed immersions are preserved by fpqc descent using the idea of Grothendieck's EGA.
For a starter, I would like to prove the title theorem. We need some definitions to state it.
Definition 1 We say a continuous map $f\colon X \rightarrow Y$ of topological spaces is quasi-compact if $f^{-1}(U)$ is quasi-compact for every quasi-comapact open subset $U$ of $Y$. We say a subset $Z$ of a topological space $X$ is retrocompact if the canonical injection $Z \rightarrow X$ is quasi-compact.
Definition 2 Let $X$ be a topological space. A subset $Z$ of $X$ is called constructible if it is a finite union of subsets of the form $U - V$, where $U, V$ are retrocompact open subsets of $X$.
Definition 3 We say a subset $Z$ of a topological space $X$ is locally constructible in $X$ if for every point $x$ of $X$ there exists an open neighborhood $V$ of $x$ such that $Z \cap V$ is constructible in $V$.
Definition 4 Let $f\colon X \rightarrow Y$ be a morphism of schemes. We say $f$ is quasi-separated if the diagonal morphism $\Delta\colon X \rightarrow X\times_Y X$ is quasi-compact.
Definition 5 Let $B$ be an algebra over a ring $A$. Suppose there exists a surjective $A$-homomorphism $\psi\colon A[X_1, \cdots, X_n] \rightarrow B$, where $A[X_1, \cdots, X_n]$ is a polynomial ring. If Ker $\psi$ is a finitely generated ideal, we say $B$ is finitely presented over $A$ or $B$ is a finitetely presented $A$-algebra.
Definition 6 Let $f\colon X \rightarrow Y$ be a morphism of schemes. Let $x \in X$. If there exists an open neighborhood $U$ of $x$ and an open neighborhood $V$ of $f(y)$ such that $f(U) \subset V$ and $\mathcal {O}_X(U)$ is finitely presented over $\mathcal{O}_Y(V)$, then we say $f$ is locally of finite presentation at $x$. We say $f$ is locally of finite presentation if it is locally of finite presentation at every point of $X$.
Definition 7 Let $f\colon X \rightarrow Y$ be a morphism of schemes. If $f$ is locally of finite presentation, quasi-compact and quasi-separated, we say $f$ is of finite presentation.
Theorem of Chevalley Let $f\colon X \rightarrow Y$ be a morphism of finite presentation. For any locally constructible subset $Z$ of $X$, $f(Z)$ is locally constructible in $Y$.
My question How do you prove it?
Remark Though I plan to post a proof of this theorem, I welcome anyone's proof. There are usually several different solutions to a mathematical question each of which may have its own merit.