Timeline for Holomorphic vector bundles and Swan's theorem
Current License: CC BY-SA 2.5
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| Aug 22, 2010 at 13:48 | comment | added | Donu Arapura | I was also curious, so I looked up the paper. My German is very poor, so I may be misunderstanding, but it seems in Satz 6.2, Forster proves directly that locally free implies projective, by first getting a surjection $\mathcal{O}_X^n\to \mathcal{E}$ etc. as you say. However, in any case, the original question has a positive answer either by Forster, or by the arguments outlined in the comments below. | |
| Aug 22, 2010 at 5:58 | comment | added | BCnrd | I haven't seen Forster's paper, but are you sure the final sentence in this answer isn't circular reasoning? I would guess (since it is the natural argument coming to mind) that the way Forster proves the global sections of a vector bundle are a projective module is by proving it's a direct summand of a finite free module, and that in turn is proved by exhibiting the vector bundle as a direct summand of a finite free vector bundle (using Steinness). In the end you are getting a correct deduction, but it feels like the black box may rest on what you are deducing from it. Did you check? | |
| Aug 21, 2010 at 19:54 | vote | accept | Vamsi | ||
| Aug 21, 2010 at 19:02 | history | answered | Daniel Pomerleano | CC BY-SA 2.5 |