$\def\sO{\mathcal{O}}\def\sF{\mathcal{F}}$On the Stacks Project there are several instances where the seemingly undefined notion of a “morphism of schemes that induces a bijection between irreducible components” is invoked, see 037A, 037O, 01RO, 035L, 035Q; an internal search throws even more instances.
My question is:
What is the definition of this concept?
I guess J. de Jong is the person that can ultimately answer this question, but I am asking here since maybe the notion is explicitly defined in other sources (knowing which would help too).
Until today, I thought the definition was:
Definition 1. We say that a continuous map $f:X\to Y$ of topological spaces induces a bijection between irreducible components if for all irreducible components $Z\subset X$, it holds that $\overline{f(Z)}$ is an irreducible component and the map $Z\mapsto \overline{f(Z)}$ is a bijection between sets of irreducible components.
This notion is local on the target, stable under composition and it satisfies that if $g\circ f$ and $f$ have the property, then so does $g$.
The situation for schemes is the expected one:
Lemma 2. Let $f:X\to Y$ be a continuous map between sober topological spaces. Then $f$ induces a bijection between irreducible components (in the sense of Definition 1) if and only if for every generic point $\xi\in X$ of an irreducible component of $X$, $f(\xi)$ is a generic point of an irreducible component of $Y$ and $\xi\mapsto f(\xi)$ is a bijection between sets of generic points of irreducible components.
(The proof is because $f(x)=y$ iff $\overline{f\left(\overline{\{x\}}\right)}=\overline{\{y\}}$, for $X$, $Y$ are sober.) I said “until today” because so far I worked on my mind with this definition when reading SP. However, I just noticed that 01RO references:
EGA I, (2.2.9) Soient $(X,\sO_X)$, $(Y,\sO_Y)$ deux préschémas; on suppose que $X$ et $Y$ ont un même nombre fini de composantes irréductibles $X_i$ (resp. $Y_i$) ($1\leq i\leq n$); soit $\xi_i$ (resp., $\eta_i$) le point générique de $X_i$ (resp. $Y_i$). On dit qu'un morphisme $$ f=(\psi,\theta):(X,\sO_X)\to(Y,\sO_Y) $$ est birationnel si, pour tout $i$, $\psi^{-1}(\eta_i)=\{\xi_i\}$ et $\theta_{\xi_i}^\sharp:\sO_{\eta_i}\to\sO_{\xi_i}$ est un isomorphisme.
From this, one could argue that what SP calls “inducing a bijection on irreducible components” means “one has Definition 1 AND $f^{-1}(\eta)$ is a singleton for every generic point $\eta\in Y$ of an irreducible component.” This is a strictly stronger condition (consider a morphism $X\to \operatorname{Spec}k$ with $X$ irreducible and with more than one point).
What is the correct definition then? Maybe my definition and EGA's implicit one are equivalent in the case from Grothendieck's text quoted above? Explicitly:
Problem. Let $f:X\to Y$ be a morphism of schemes with finitely many irreducible components such that for every generic point $\xi\in X$ of an irreducible component, $\sO_{Y,f(\xi)}\to\sO_{X,\xi}$ is an iso. Then $f:X\to Y$ satisfies Definition 1 above implies that $f^{-1}(\eta)$ is a singleton for every generic point $\eta\in Y$ of an irreducible component.
This is what I thought for a proof: Let $x\in f^{-1}(\eta)$. Let $\xi\in X$ be the generic point of an irreducible component passing through $x$; in particular, $\xi\rightsquigarrow x$. Hence, $f(\xi)\rightsquigarrow f(x)=\eta$. By minimality, $f(\xi)=\eta$. Thus, we have a commutative diagram
(the map $\sO_{X,x}\to\sO_{X,\xi}$ is defined in the following way: if $\sF$ is a sheaf over a topological space $A$ and $a,b\in A$ are s.t. $a\rightsquigarrow b$, then there is a natural map $\sF_b\to\sF_a$.)
Since the top map is an iso (by hypothesis), then the left and right maps are injective and onto, respectively. But here's when I get stuck; I don't see how to go any further.
