Timeline for Stable vector bundle Projective or Injective?
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2015 at 16:32 | comment | added | Francesco Polizzi | @abx: right, thanks. I was confused because I was thinking about the Serre-Swan theorem, but it actually holds for affine varieties. | |
| Feb 9, 2015 at 16:18 | comment | added | abx | @Francesco : just take the standard definition of injective or projective (the usual lifting property) in the category of $\mathscr{O}$-modules; this is equivalent to the vanishing of global Ext. | |
| Feb 9, 2015 at 15:39 | comment | added | Leo Alonso | The Ext sheaves correspond to a property we may call "internal projectivity" (exactness of ext-sheaves in the first variable), which is indeed a property that holds for vector bundles on schemes. | |
| Feb 9, 2015 at 15:33 | comment | added | Francesco Polizzi | I'm confused. You should take the Ext sheaves, not the Ext groups, or I'm mistaken? As far as I remember, a finitely generated module is locally free if and only it is projective. So a coherent, locally free $\mathcal{O}_X$-module should be projective. I'm missing something? | |
| Feb 9, 2015 at 15:27 | vote | accept | Z.A.Z.Z | ||
| Feb 9, 2015 at 15:25 | history | answered | abx | CC BY-SA 3.0 |