Timeline for Resolving complexes of coherent analytic sheaves
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Dec 13, 2022 at 23:23 | comment | added | Tim | O'Brian–Toledo–Tong's work is also where I learnt basically everything I know about Stein spaces, but it's true that they often just state things without mentioning any proofs... but the Forster article seems like a good reference (shockingly I couldn't find this in Grauert and Remmert's "Theory of Stein Spaces", though maybe I just wasn't looking in the right places) | |
| Dec 13, 2022 at 20:21 | comment | added | Richard Lärkäng | It seems indeed also to be proven in one direction of Satz 6.2 in the article of Forster. From a quick look, the key step in that proof seems to be Satz 2.1, which I believe shows that the global sections functor establishes an equivalence of categories between coherent sheaves on a Stein space and so-called "Stein modules". Then, Satz 6.2 is proven simply by the fact that sheaf $\mathcal{H}om(P,\bullet)$ is exact if P is a vector bundle, and taking global sections $\Gamma(X,\bullet)$ is exact on the category of coherent sheaves on a Stein space. | |
| Dec 13, 2022 at 20:20 | comment | added | Richard Lärkäng | I am not aware of the history of the statement that vector bundles over Stein spaces are projective. I first learnt about it in O'Brian-Toledo-Tong, "The trace map and characteristic classes for coherent sheaves", where it is mentioned in passing after Lemma 1.6 without any indication of proof. | |
| Dec 13, 2022 at 20:18 | comment | added | Richard Lärkäng | Indeed, the argument as written uses the useful fact that vector bundles over Stein spaces are projective, although I suppose since you anyway need to shrink the neighborhood when obtaining the resolutions, by further shrinking the neighborhood, you could probably make do with the more elementary fact that trivial vector bundles are projective. | |
| Dec 13, 2022 at 15:02 | vote | accept | Tim | ||
| Dec 13, 2022 at 15:02 | comment | added | Tim | Thank you! So it seems like the key ingredient which lets you apply "classical" constructions is exactly this fact that vector bundles are projective over Stein spaces. I think a reference for this fact is Forster's "Zur Theorie der Steinschen Algebren und Moduln", but my German is not good enough to check... but like you say, it should follow from the local to global Ext spectral sequence. | |
| Dec 7, 2022 at 20:50 | history | answered | Richard Lärkäng | CC BY-SA 4.0 |