Timeline for Functorial lift of certain vector bundles to the ambient projective space
Current License: CC BY-SA 4.0
12 events
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| Feb 13, 2021 at 13:30 | history | edited | user127776 | CC BY-SA 4.0 |
Added another relevant question.
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| Feb 13, 2021 at 2:53 | comment | added | Ben C | @user127776 some people require that morphisms of vector bundles have constant rank on the fibers such that the kernel and cokernel are also vector bundles. I will assume from your comment that you do not. | |
| Feb 13, 2021 at 1:59 | history | edited | user127776 | CC BY-SA 4.0 |
added 71 characters in body
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| Feb 13, 2021 at 1:58 | comment | added | user127776 | @Ben C I am not sure I understand your question correctly, but they need to be morphism of coherent sheaves or $\mathcal{O}_X$ modules. | |
| Feb 13, 2021 at 1:54 | comment | added | Ben C | Do your morphisms have to have constant rank or are arbitrary sheaf maps allowed? | |
| Feb 12, 2021 at 23:20 | history | edited | user127776 | CC BY-SA 4.0 |
Added extra question.
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| Feb 12, 2021 at 15:58 | history | edited | user127776 | CC BY-SA 4.0 |
added 223 characters in body
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| Jan 10, 2021 at 1:17 | comment | added | user127776 | You are right, but isn't it true that at least you can lift on some neighborhood of the curve? | |
| Jan 10, 2021 at 0:43 | comment | added | dhy | Indeed, I misread the question and thought you were lifting the category of all bundles. But now I think there is a different issue. You write "the same is true for the morphisms..." but that amounts to choosing a section of $\Gamma(\mathcal{O}(m-n))\rightarrow\Gamma_C(\mathcal{O}(m-n)).$ This map is not surjective in general so I am skeptical of the claim that there is always a lifting of the morphisms. Am I missing something? | |
| Jan 10, 2021 at 0:05 | comment | added | user127776 | I think you forgot about the powers of $L$. That is the category I am asking. There is no problem in lifting the bundles themselves, all of them are lift-able by the definition. | |
| Jan 9, 2021 at 23:39 | comment | added | dhy | Unfortunately this is not possible in general: The only bundles you can get from restrictions from the ambient projective space have K-theory classes a linear combination of the classes of $\mathcal{O}$ and $L$. | |
| Jan 9, 2021 at 23:24 | history | asked | user127776 | CC BY-SA 4.0 |