Let $j: X \to Y$ be a closed embedding. Let $I_{X/Y}$ be the ideal sheaf of this closed embedding. Then there is a exact sequence

$$ I_{X/Y} \to \mathcal{O}_Y \to j_{*}\mathcal{O}_X \to 0$$

One use this exact sequence for computation of $H^i(Y, I_{X/Y})$.

In general it is easy to compute $H^i(Y, \mathcal{O}_Y)$ and $H^i(Y, j_{*}\mathcal{O}_X)$. But how does one determine the maps between $H^i(Y, \mathcal{O}_Y)$ and $H^i(Y, j_{*}\mathcal{O}_X)$? Are there any references where one can find examples of this kind of computation?