Timeline for When is the flatness locus non-empty
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 21, 2015 at 20:11 | vote | accept | user46578 | ||
| Oct 21, 2015 at 20:08 | comment | added | Ariyan Javanpeykar | @user46578 An open set could be empty. (In our situation, the morphism $f$ is generically flat and of finite presentation. The locus of flatness is therefore open (by Thm 11.3.1) and non-empty (by generic flatness). In particular, as $S$ is integral, the locus of flatness is dense open in $S$.) | |
| Oct 21, 2015 at 20:06 | history | edited | Ariyan Javanpeykar | CC BY-SA 3.0 |
added 151 characters in body
|
| Oct 21, 2015 at 20:06 | comment | added | user46578 | @LaurentMoret-Bailly But theorem $11.3.1$ does not mention non-emptyness of the flat locus. | |
| Oct 21, 2015 at 19:58 | comment | added | Ariyan Javanpeykar | @LaurentMoret-Bailly Yes, thank you! (Thus, I should have written, "any morphism of schemes $f:X\to S$ with $S$ integral is generically flat. If $f$ is additionally of finite presentation, then $f$ is flat over a dense open of $S$.") | |
| Oct 21, 2015 at 19:51 | comment | added | Laurent Moret-Bailly | @AriyanJavanpeykar : you probably need $f$ to be of finite presentation. In fact if $f$ is locally of finite presentation, then the flat locus in $X$ is open (EGA IV, (11.3.1)). If in addition $f$ is quasi-compact, your claim follows. | |
| Oct 21, 2015 at 19:37 | comment | added | Ariyan Javanpeykar | Any morphism of schemes $f:X\to S$ with $S$ integral is flat over some dense open of $S$. As Matthieu Romagny mentions above, the question is more interesting when $S$ is only irreducible, e.g., $S$ is Spec $k[x]/x^n$. | |
| Oct 21, 2015 at 19:34 | comment | added | Ariyan Javanpeykar | Right. "Everything is flat over a field." | |
| Oct 21, 2015 at 19:32 | comment | added | user46578 | Wow. So I do not even need locally of finite-presentation or proper morphism, right? | |
| Oct 21, 2015 at 19:29 | history | answered | Ariyan Javanpeykar | CC BY-SA 3.0 |