Computation of cohomology of ideal sheaves

Question

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?

Answer 1

The computation of the restriction morphism $H^i(Y,O_Y) \to H^i(X,O_X)$ is usually non-trivial (unless you know for instance an explicit locally free resolution of the ideal sheaf of $X$ in $Y$).

On $H^0$ this is quite easy, of course.

For $H^1$ one can act as follows: each class in $H^1(Y,O_Y)$ can be represented as an extension $$ 0 \to O_Y \to E \to O_Y \to 0; $$ restricting it to $X$ one obtains an exact sequence $$ 0 \to O_X \to E\vert_X \to O_X \to 0; $$ the restriction map takes the class of the first to the class of the second.

On higher $H^i$ one can use Yoneda representations too, but it becomes more complicated.