Timeline for flatness of coherent analytic sheaf
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 19, 2011 at 13:37 | comment | added | shenghao | Similarly, a coherent sheaf $F$ on a complex manifold $X$ is flat if and only if for every $x\in X,$ the stalk $F_x$ is a flat $O_x$-module. As $O_x$ is Noetherian local (cf. Gunning's books), flat=free. And $F_x$ free over $O_x$ implies that $F$ is locally free (see Hartshorne II ex. 5.7 for an algebraic counterpart, and mimic its proof). So a coherent sheaf $F$ being flat if and only if it's locally free (I could be wrong though...). | |
| May 19, 2011 at 6:13 | comment | added | HKSHLZW | thank you very much ! And how to determine the flatness of a giving coherent sheaf in general , does there exist some kind of obstruction ? | |
| May 18, 2011 at 13:38 | comment | added | shenghao | To check that locally free coherent sheaves are flat (as answered by Ottem), one can pass to stalks. | |
| May 18, 2011 at 13:01 | answer | added | J.C. Ottem | timeline score: 5 | |
| May 18, 2011 at 12:49 | history | asked | HKSHLZW | CC BY-SA 3.0 |