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Let $i:Z\to X$ be a closed embedding of (projective) varieties; $S$ is a coherent sheaf on $X$. How could one compute $H^*(Z,i^\ast S)$ (I don't know whether I should write $H^\ast (Z,i^{-1}S)$ instead? If there is a substantial distinction, I could be interested in both)? Could one do this using some infinitesimal neighbourhoods of $Z$ in $X$?

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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. – Donu Arapura Oct 17 '11 at 22:12
Could you say more about resolutions? For which $S$ the computation of $H^\ast(Z,i^*S)$ is easy?! Thank you! – Mikhail Bondarko Oct 18 '11 at 3:22
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. – Donu Arapura Oct 18 '11 at 11:46

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