Timeline for Inverse Function Theorem in Algebraic Geometry
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 26, 2013 at 20:55 | comment | added | Tom Goodwillie | In some trivial but profound sense every etale map is locally an isomorphism. Let "locally" mean "locally in the etale topology". | |
| May 26, 2013 at 18:46 | vote | accept | Peter Crooks | ||
| May 26, 2013 at 18:10 | answer | added | Sean Lawton | timeline score: 2 | |
| May 26, 2013 at 6:52 | comment | added | Jérémy Blanc | If $X=Y=\mathbb{A}^n$, your question is the famous Jacobian conjecture. | |
| May 25, 2013 at 23:43 | comment | added | David Roberts♦ | What about something of the form: "every smooth map is locally split over an étale cover"? I vaguely remember reading this somewhere, can't be sure it's true. | |
| May 25, 2013 at 23:17 | comment | added | Qfwfq | You can only say $f$ is a local biholomorphism around $x$. It is not, in general, a local isomorphism of algebraic varieties around $x$: just take a covering of curves of different genus minus the ramification points. Or, even simplier, take your example $\mathbb{A}^1\setminus \{ 0 \} \to \mathbb{A}^1\setminus \{ 0 \}$ $z\mapsto z^2$. The inverse function theorem however holds with respect to the étale topology for étale morphisms. | |
| May 25, 2013 at 22:41 | history | asked | Peter Crooks | CC BY-SA 3.0 |