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I have a problem with a detail of Qing Liu's proof of Proposition 4.3.8 (pag. 137 of "Algebraic Geometry and Arithmetic Curves"). The statement is:

Let $Y$ be a scheme having only a finite number of irreducible components. Let $f:X\rightarrow Y$ be a flat morphism. Let us suppose that $Y$ is reduced (resp. irreducible; resp. integral) and that the generic fibers of $Y$ are also reduced (resp. irreducible; resp. integral); then $X$ is reduced (resp. irreducible; resp. integral).

The proof starts with irreducibility:

Let us suppose $Y$ is irreducible with generic point $\eta$, and $X_{\eta}$ irreducible. Then $Z:=\overline{X_{\eta}}$ is a closed irreducible subset of $X$. Its complement $U=X\setminus Z$ is an open subset of $X$ which does not dominate $Y$.

My problem is that I cannot find a good reason for this last adfirmation. I could fix it just in the case of $X$ with a finite number of irreducible components, but how can I prove it for the general case? Thank you :)

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  • $\begingroup$ In the last line of question, how exactly is "finiteness of irreducible components" being used? The way I see the proposition(with irreducible hypothesis) and its proof, is that there is a exactly one irreducible closed subset $X' \subset X$ which dominates Y and so $U=X-X'$ is $\emptyset$ due to the flatness assumption. $\endgroup$
    – sriram
    Commented Dec 19, 2023 at 10:56
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    $\begingroup$ The argument works as stated if $f$ is finitely presented, because then $f(U)$ is open. In general, just note that $f_{\mid U}$ is flat, hence generalizing (this is just the ``going-down theorem''). In particular if $f(U)$ is not empty it contains $\eta$, a contradiction. (I am not sure we can conclude that $f(U)$ is not dense.) $\endgroup$
    – Laurent Moret-Bailly
    Commented Dec 19, 2023 at 18:51
  • $\begingroup$ @LaurentMoret-Bailly Thank you, I didn't know the "going down" for flat ring morphisms, but now I think the argument is clear since we can work on affine opens and $\eta$ is minimal (so it is in the image of $U$ for the g-down theorem and this is absurd). $\endgroup$
    – BernyPiffaro
    Commented Dec 19, 2023 at 21:58
  • $\begingroup$ I was wondering that in Lemma 4.3.7 which precedes the proposition in question, there doesn't seem to be any finite presentation assumption on the morphism $f$ and it concludes only the weaker statement that $f(U)$ is dense in Y(in contrast with $f(U)$ is open in Y) which is all we require for the conclusion of 4.3.8. Qing Liu doesn't seem to have finitely presentation assumption in the definition of flat ring homomorphisms as well. So may be one can conclude without finite presentation hypothesis? $\endgroup$
    – sriram
    Commented Dec 20, 2023 at 9:35
  • $\begingroup$ @sriram Yes, there isn't the finitely presentation assumption in the defn of flat morphism. Indeed I think I solved my problem using just the "going down" for flat morphisms and working on affine opens. $\endgroup$
    – BernyPiffaro
    Commented Dec 21, 2023 at 10:32

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