Timeline for Can non-split extension be isomorphic to the split one as objects
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 17, 2020 at 20:37 | comment | added | Francesco Polizzi | @NicolasHemelsoet: right, in fact I was assuming that some non-zero element of $H^0(\mathcal{O} \oplus \mathcal{O}(n))$ would give a non-split sequence, but I did not check details. | |
| Nov 17, 2020 at 20:11 | vote | accept | user127776 | ||
| Nov 17, 2020 at 20:06 | comment | added | Nicolas Hemelsoet | @FrancescoPolizzi : but for example, there is a non-split sequence $0 \to \mathcal O \to \mathcal O(1)^{\oplus 2} \to \mathcal O(2) \to 0$ corresponding to a non-zero element in $Ext^1(\mathcal O(2), \mathcal O)$. So given a non-zero class in $Ext^1(\mathcal O, \mathcal O(n))$ do you know that the middle term will be given by $\mathcal O \oplus \mathcal O(n)$ ? | |
| Nov 17, 2020 at 20:00 | answer | added | SashaP | timeline score: 11 | |
| Nov 17, 2020 at 19:26 | comment | added | Francesco Polizzi | You can take a non-split short exact sequence on $\mathbb{P}^1$ of the form $$ 0 \to \mathcal{O} \to \mathcal{O} \oplus \mathcal{O}(n) \to \mathcal{O}(n) \to 0.$$ It exists as soon as $n \geq 2$, as one can check by computing the Ext$^1$ group. | |
| Nov 17, 2020 at 19:24 | comment | added | LSpice | See also mathoverflow.net/questions/163041/… . | |
| Nov 17, 2020 at 18:50 | history | edited | user127776 | CC BY-SA 4.0 |
added 7 characters in body
|
| Nov 17, 2020 at 18:50 | comment | added | user127776 | As vector bundles. Not as extensions (which obviously is impossible according to the assumptions). | |
| Nov 17, 2020 at 18:49 | comment | added | Jef | What do you mean by 'as objects'? | |
| Nov 17, 2020 at 18:40 | history | asked | user127776 | CC BY-SA 4.0 |