Timeline for Regular monomorphisms of schemes
Current License: CC BY-SA 3.0
10 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jun 5, 2011 at 17:02 | comment | added | Martin Brandenburg | But I still don't understand it. In general, if we have morphisms $S \to X,Y$ of topological spaces, then the open subsets of $X \cup_S Y$ correspond bijectively to pairs $U,V$ of open subsets of $X,Y$, such that their preimage in $S$ coincide (this follows by appliying the universal property to the Sierpinski space, but also an explicit construction reveals this). If we have such a pair $U,V$ with preimage $T$ in $S$, then the open subset in $X \cup_S Y$ is identified with $U \cup_T V$. But in your situation, it is not possible to write $X \cup_Z U$ as such an open subset of $X \cup_W X$. | |
| Jun 5, 2011 at 16:51 | comment | added | Martin Brandenburg | The answer by Laurent Moret-Bailly indicates that your idea is correct if we have the stronger assumption that $Z \to X$ is an open immersion $Z \to W$ followed by an closed immersion $W \to X$; then we can also factor it as a closed immersion $Z \to U$ followed by an open immersion $U \to X$ (all this is trivial in the category of topological spaces, where also the converse is correct, but in the category of schemes extra care has to be taken). | |
| Jun 5, 2011 at 7:29 | answer | added | Laurent Moret-Bailly | timeline score: 2 | |
| Jun 3, 2011 at 16:02 | comment | added | Tom Goodwillie | Let $V$ be the complement of $W$ in $X$. Couldn't you make the pushout $X\cup_ZU$ by gluing together $V$ and $U\cup_ZU$ along the common open subscheme $V\cap U$ (which is also the complement of $Z$ in $U$, so can be identified with the complement of (the second copy of) $U$ in $U\cup_ZU$)? Maybe I'm missing some subtle point. | |
| Jun 3, 2011 at 15:35 | comment | added | Tom Goodwillie | I'm not an algebraic geometer, I'm just guessing. You said "in general we have a composition of a closed immersion followed by an open immersion", so I thought that if $Z$ is locally closed in $X$ then that makes $Z$ a closed subscheme of an open subscheme $U$ of $X$. | |
| Jun 3, 2011 at 14:22 | comment | added | Martin Brandenburg | Which scheme structure do you chooce on $Z$? / The pushouts $X \cup_Z U, U \cup_Z X$ do not have to exist. | |
| Jun 3, 2011 at 13:27 | comment | added | Tom Goodwillie | What happens if you combine the two constructions like this? If $U$ is open in $X$ and $Z=U\cap W$ is closed in $U$, where $W$ is closed in $X$, then first make $X\cup_Z U$ and $U\cup_Z X$ (which are open in $X\cup_WX$), then glue them along $U\cup_Z U$ which is open in both of them. I haven't checked that this makes sense. | |
| Jun 3, 2011 at 12:22 | history | edited | Martin Brandenburg | CC BY-SA 3.0 |
added 82 characters in body
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| May 28, 2011 at 14:10 | history | asked | Martin Brandenburg | CC BY-SA 3.0 |