Timeline for morphism which is open but not universally open
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 22, 2010 at 14:42 | comment | added | Laurent Moret-Bailly | @Martin: since you mention it, is it open? | |
| Oct 20, 2010 at 9:48 | comment | added | Martin Brandenburg | Is $Spec(\hat{A}) \to Spec(A)$ universally open? | |
| Oct 20, 2010 at 4:45 | comment | added | BCnrd | Dear Tom: yes (just like for closedness of a morphism). | |
| Oct 19, 2010 at 16:39 | comment | added | Tom Goodwillie | Does "open" for a scheme morphism simply mean open as a map of topological spaces? | |
| Oct 19, 2010 at 16:27 | comment | added | Laurent Moret-Bailly | This example is the "standard" one among morphisms of finite type between notherian schemes. In fact, the failure of an open morphism to be universally open is related to the non-normality of the base: see EGA IV.3 (14.4.9). | |
| Oct 19, 2010 at 15:25 | comment | added | user565739 | To Moret-Bailly, you mean the example in EGA? I just took it a look . Just wanna know if there is another example which looks eaiser. | |
| Oct 19, 2010 at 15:02 | comment | added | Laurent Moret-Bailly | What is wrong with that example? | |
| Oct 19, 2010 at 14:36 | comment | added | user565739 | Thank you, BCnrd. I would appretiate that anyone gives an example which is open but not universally open, an example different from the one in EGA IV.3, Remarque 14.3.9.i . | |
| Oct 19, 2010 at 13:44 | comment | added | BCnrd | The map $i'$ is open. Indeed, as you note, the only way it can fail to be open is if the diagonal closed point in the source is also open. But if such openness holds, this point would split off as a clopen subscheme (with suitable 1-point scheme structure) which is then necessarily flat over the base since $i'$ is flat. But that's absurd, since it would be a 1-point scheme lying over the closed point of the base and hence has uniformizer on the base pulling back to a nilpotent function, contradicting torsion-freeness over the base which is a consequence of flatness. | |
| Oct 19, 2010 at 12:20 | history | asked | user565739 | CC BY-SA 2.5 |