Section 5.6 of EGAII is dedicated to Chow's lemma. I am having a hard time following an early step of the proof.
The version of Chow's lemma in the text assumes that $X$ is a separated $S$-scheme of finite type where $S$ is either noetherian or else more generally that $S$ is quasi-compact and $X$ has a finite number of irreducible components. In these cases it is conlcuded that
(i) there exists a quasi-projective $S$-scheme and a surjective, projective morphism $f:X'\to X$,
(ii) that there is an open $U\subseteq X$ such that $f|_{f^{-1}U}:f^{-1}(U)\to U$ is an isomorphism.
The step I am confused about is the one when the proof first reduces to the case when $X$ is irreducible. To do this it says to assumes that Chow's lemma was proved in the irreducible case, and thus taking $X_i$ to be the irreducible components, and $f_i:X'_i\to X_i$ satisfying the conditions of Chow's lemma, we have that if $X'$ is the disjoint union of $X_i'$ and $f$ the morphism from $X'\to X$ which restricts to $f_i$ on $X_i'$, then $f:X'\to X$ is a witness to Chow's lemma.
However, what confuses me here is how we can take the individual $X_i$. What is the scheme structure on them? If we take the reduced structure (which is what the proof seems to be saying to do), condition (ii) will not necessarily hold? Is there a way to fix this?
I notice that the Stacks project goes with a different approach for the proof and they seem to need that $S$ is actually noetherian. Is this way to fix it?
Thanks for any help
I think the following works. In the noetherian case, you can just take $V_i = U_i$ below.
Let $S$ be qcqs, let $X \to S$ be separated and of finite type, and let $X = X_1 \cup \cdots \cup X_r$ be a decomposition of $X$ into finitely many irreducible components. Let $U_i = X \setminus \bigcup_{i \ne j} X_j$; note this is a non-empty open subset of $X$ contained in $X_i$ that does not meet any other components, and that $U_i$ contains the generic point $\eta_i$ of $X_i$. Further, let $V_i$ be an affine open subset of $U_i$ containing the generic point $\eta_i \in X_i$. Then, $V_i$ is open in $X$, hence has a canonical open subscheme structure, and $V_i \hookrightarrow X$ is quasi-compact (using the quasi-separatedness of $X$). By [Stacks, Tag 01R8], the scheme-theoretic closure of $V_i$ is therefore a closed subscheme of $X$ with underlying topological space $X_i$.