Timeline for Ideals of etale structure sheaves
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 27, 2012 at 11:18 | vote | accept | Colin McLarty | ||
| Jul 26, 2012 at 1:21 | answer | added | Colin McLarty | timeline score: 3 | |
| Jul 19, 2012 at 11:51 | comment | added | Damian Rössler | @Colin McLarty. You are right about the ideal sheaves of course... sorry for the silly question. | |
| Jul 18, 2012 at 15:41 | comment | added | Yosemite Sam | OK, got it. thanks for clearing that up. | |
| Jul 18, 2012 at 15:24 | comment | added | Colin McLarty | More fully: Does every sheaf of ideals $\mathscr{I}\subseteq \mathscr{O}_{\mbox{et}X}$ of the etale structure sheaf of a Noetherian scheme have a finite set of local sections such that every section is covered by a linear combination of restrictions of those? This reduces to: For a Noetherian ring $R$ and etale sheaf of ideals $\mathscr{I}$ on $R$ is there a faithfully flat etale ring extension $R\rightarrow S$ such that for every etale $S\rightarrow T$ every section in $\mathscr{I}(T)$ is a linear combination of restrictions of sections in $\mathscr{I}(S)$? | |
| Jul 18, 2012 at 14:34 | comment | added | Colin McLarty | No. I don't think so. Over the spectrum of the integers consider the ideal which is 0 over every open including (2) and is the unit ideal over every open which does not include (2). | |
| Jul 18, 2012 at 7:54 | comment | added | Damian Rössler | Isn't every sheaf of ideals quasi-coherent, if the scheme is noetherian (and thus every étale open also is) ? | |
| Jul 17, 2012 at 23:11 | comment | added | Colin McLarty | But I need not only quasi-coherent (sheaves of) ideals. I want to know this for all (sheaves of) ideals. | |
| Jul 17, 2012 at 22:44 | comment | added | Yosemite Sam | OK guys, I'm being really really thick right now. What's the difference with the Zariski structure sheaf? Aren't the two categories (etale/zariski quasi-coherent modules) equivalent for schemes (via the forget morphism)? | |
| Jul 17, 2012 at 19:17 | comment | added | Colin McLarty | Yes, though I would write $U\mapsto \mathscr{O}_U(U)$ to define it. | |
| Jul 17, 2012 at 18:01 | comment | added | Martin Brandenburg | The sheaf $U \mapsto \mathcal{O}_X(U)$ defined on étale opens $U \to X$. | |
| Jul 17, 2012 at 16:00 | comment | added | Yosemite Sam | what is the etale structure sheaf? | |
| Jul 17, 2012 at 15:58 | history | asked | Colin McLarty | CC BY-SA 3.0 |