Timeline for Do infinitesimal neighbourhoods help to compute the inverse images of coherent sheaves?
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| Oct 18, 2011 at 11:46 | comment | added | Donu Arapura | For example, if $X$ is projective space then Hilbert's syzygy theorem would allow you to resolve the ideal of $Z$ by graded free modules. This means that the ideal sheaf could be resolved by sums of line bundles. You could compute cohomology of $H^*(Z,\mathcal{O}_Z)$ using this. | |
| Oct 18, 2011 at 3:22 | comment | added | Mikhail Bondarko | Could you say more about resolutions? For which $S$ the computation of $H^\ast(Z,i^*S)$ is easy?! Thank you! | |
| Oct 17, 2011 at 22:12 | comment | added | Donu Arapura | Take $Z=x$ to be a point and $S=\mathcal{O}_X$, then $H^0(i^*S)=k$ and $H^0(i^{-1}S)=\mathcal{O}_x$. So they are certainly different. Computing $H^∗(Z,i^∗S)$ is not so easy in general, and usually it is more useful to resolve the ideal sheaf than to play with inf. nbhds I'm not quite sure what else to say. | |
| Oct 17, 2011 at 19:52 | history | asked | Mikhail Bondarko | CC BY-SA 3.0 |