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 $X$?
