Timeline for Extending etale morphisms
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2020 at 10:32 | comment | added | user45397 | @LaurentMoret-Bailly and Anonymous; Thank you for the answer | |
| Jul 12, 2020 at 7:29 | comment | added | Laurent Moret-Bailly | In this case the construction is very simple: take $g$ as in Anonymous's comment. Then $\ol{X}=\mathrm{Spec}(\mathscr{A}))$ where $\mathscr{A}\subset g_*\mathscr{O}_Z$ is the subalgebra of sections whose restriction on $g^{-1}(y)$ is constant (i.e. comes from $\kappa(y)$). | |
| Jul 12, 2020 at 7:19 | comment | added | Laurent Moret-Bailly | For the existence of the crushing (also called "pinching", or Ferrand pushout) see this paper, or this section of the Stacks Project. | |
| Jul 12, 2020 at 5:55 | comment | added | user45397 | @Anonymous Thank you. Could you put this in the answer. Especially, could you elaborate a little (or give reference) for what you mean by "crushing". Is it something like blowing down? If so, why does a "crushing" always exist? I think the answer lies in what you say "suitable fibre product at the level of rings", but this statement is not clear to me. | |
| Jul 12, 2020 at 3:13 | comment | added | Anonymous | Yes: first choose any finite morphism $g:Z \to Y$ extending $f$ with $Z$ integral (e.g., by normalizing $Y$ in $X$), and then construct $\overline{X}$ by crushing the finite closed subscheme $g^{-1}(y) \subset Z$ to a single point (so take a suitable fibre product at the level of rings). | |
| Jul 11, 2020 at 21:22 | history | asked | user45397 | CC BY-SA 4.0 |