# Computation of cohomology of ideal sheaves

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?

• Are you aware that $H^i(Y,j_*\mathcal{O}_X)$ is nothing but $H^i(X,\mathcal{O}_X)$?? – abx Nov 1 '18 at 5:00
• Yes, I'm aware of that. – john Nov 1 '18 at 13:43
• If $Y$ and $X$ are smooth and projective, you can use Hodge theory to identify your morphism to the (conjugate of) the restriction $H^0(Y,\Omega ^i_Y)\rightarrow H^0(X,\Omega ^i_X)$, which may be easier to analyze. – abx Nov 1 '18 at 19:49
• The precise description of the maximal rank conjecture for curves sheds light on why this may be hard, even though the context is a little different (see arxiv.org/pdf/1711.04906.pdf for a statement of the conjecture). – Anwesh Ray Nov 1 '18 at 19:59

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.