I am trying to understand 0CDV from the Stacks Project, whose slogan is the title of this post.
The proof invokes two different results and supposedly one can deduce the result from any of them. I am interesting on how to use the second one, 035Q (3), to show the statement. The proof follows from this “because $\coprod Z_i\to X$ is integral and locally birational (as $X$ is reduced and has locally finitely many irreducible components).” If we assume $\coprod Z_i\to X$ is integral and locally birational I know how to conclude: 035Q, (3) gives a factorization $X^\nu\to\coprod Z_i\to X$, and $X^\nu\to\coprod Z_i$ is the normalization of $\coprod Z_i$. By taking the inverse image of each $Z_i$ in the latter morphism and applying 035K, the result follows.
My question is: why is $\coprod Z_i\to X$ locally birational? (I know how to see that it is integral.)
On the one hand, the morphism $\coprod Z_i\to X$ induces a bijection on irreducible components by construction. Let $Z\subset X$ be an irreducible component and let $\eta\in X$ be its generic point. Let $\mathcal{I}\subset\mathcal{O}_X$ be the ideal sheaf of sections that vanish on $Z$ (see 01J3). We must see that $\mathcal{O}_{X,\eta}\to\mathcal{O}_{\coprod Z_i,\eta}=\mathcal{O}_{Z,\eta}=\mathcal{O}_{X,\eta}/\mathcal{I}_\eta$ is an isomorphism, i.e., that $\mathcal{I}_\eta=0$. Let $U\subset X$ be a quasi-compact open neighborhood of $\eta$ (this is iff $Z$ meets $U$). Write $U=\bigcup_{\ell=1}^nU\cap Z_{i_\ell}$, where the $Z_{i_\ell}$ are the irreducible components of $X$ meeting $U$. We can suppose $Z_{i_1}=Z$. Then $F=\bigcup_{\ell=2}^n U\cap Z_{i_\ell}$ is a closed subset of $U$ and the open set $U\setminus F$ meets $U\cap Z$, i.e., $\eta\in U\setminus F$. Actually, $U\setminus F$ is a non-empty open subset contained in $U\cap Z$. Hence, $\mathcal{I}(U\setminus F)=0$, for $X$ is reduced; whence $\mathcal{I}_\eta=0$.