Timeline for Global obstructions for being a quotient of a rank $d$ vector bundle
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 5, 2020 at 19:27 | comment | added | Yosemite Stan | @R.vanDobbendeBruyn Yes! I was referring to the context of the answer not the context of the original question :) | |
| Feb 5, 2020 at 18:21 | comment | added | R. van Dobben de Bruyn | @YosemiteStan: doesn't the Serre construction mostly refer to codimension $2$ situations? Or is there a more general thing? | |
| Feb 5, 2020 at 18:19 | vote | accept | R. van Dobben de Bruyn | ||
| Jan 31, 2020 at 6:58 | comment | added | Yosemite Stan | Maybe it's worth mentioning that in many "practical" situations, by the Serre construction, the existence of an isomorphism $\text{det} N^*\cong L|_C$ is exactly what is needed to give a global resolution of $I_C$. | |
| Jan 29, 2020 at 22:54 | comment | added | R. van Dobben de Bruyn | Apologies; I misread the terms inside the $\mathscr Tor$ and thought that this statement contradicted the isomorphism $\mathscr Tor_1(\mathcal O_Z, \mathcal I_Z) \cong \wedge^2 N^*$ that you used. | |
| Jan 29, 2020 at 21:03 | comment | added | user147129 | See "Les K-groupes d'un schéma éclaté..." by Thomason (Inventiones 1993), Lemme 3.2 (the indexing is correct). | |
| Jan 29, 2020 at 19:17 | comment | added | Sasha | What is the problem with my indexing? | |
| Jan 29, 2020 at 19:03 | comment | added | R. van Dobben de Bruyn | Ah, that clarifies it (but I think your indexing might be off). | |
| Jan 29, 2020 at 16:10 | comment | added | Sasha | I don't mean that the resolution generalizes, but its consequence (the isomorphism for $Tor_i$) does. | |
| Jan 29, 2020 at 15:53 | comment | added | Sasha | In fact, it does (sometimes this is called the fundamental local isomorphism): $Tor_i(\mathcal{O}_Z,\mathcal{O}_Z) \cong \wedge^iN^*$ for any lci scheme $Z$. | |
| Jan 29, 2020 at 15:12 | comment | added | R. van Dobben de Bruyn | Could you say a word about where $\det N^*$ comes from? (I can do a $\mathscr Tor$ computation locally, but the Koszul resolution doesn't globalise by the very statement you're trying to prove.) | |
| Jan 28, 2020 at 20:33 | history | answered | Sasha | CC BY-SA 4.0 |