Timeline for Proposition 4.3.8 Qing Liu about flat morphisms of schemes
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 19 at 0:26 | comment | added | FShrike | Yes, note Liu's $4.3.7$ shows the stronger claim that the image $f(U)$ necessarily should, if $U$ is nonempty and $f$ is flat, contain the generic point $\eta$, which is here not possible. I modelled this situation topologically and the claim, just by 'domination' alone, doesn't go through (there exists a space $X$, a $T_0$ irreducible space $Y=\overline{\{\eta\}}$ for some unique $\eta$ and a continuous map $f:X\to Y$ such that $f(X\setminus\overline{f^{-1}\{\eta\}})$ is actually dense in $Y$) | |
| Dec 21, 2023 at 10:32 | comment | added | BernyPiffaro | @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. | |
| Dec 20, 2023 at 9:35 | comment | added | sriram | 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? | |
| Dec 19, 2023 at 21:58 | comment | added | BernyPiffaro | @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). | |
| Dec 19, 2023 at 18:51 | comment | added | Laurent Moret-Bailly | 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.) | |
| Dec 19, 2023 at 10:56 | comment | added | sriram | 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. | |
| Dec 18, 2023 at 20:55 | history | asked | BernyPiffaro | CC BY-SA 4.0 |