# Higher direct image and local cohomology.

Let $X$ be an scheme, $Z \subset X$ a closed subscheme, and $\mathcal{F}$ a coherent sheaf then,

$\mathcal{R}^{i-1}_{j_{*}}(\mathcal{F}|_{X-Z})\cong\mathcal{H}_{Z}^{i}(X,\mathcal{F})$

I would like to see this isomorphism explicitly. Since I dont really understand how to see the elements of $H^i_Z(X,F)$. If it is possible, how can I see them in terms of Cech Cohomology?

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The isomorphism also holds for relative cohomology of sheaves on arbitrary topological spaces. For a proof, see for example Cor. 1.9 in Hartshorne's Local cohomology (LNM 41). It is quite elementary and self-contained. For some geometric intuition for relative cohomology you may consult texts on algebraic topology (for example Hatcher's textbook), because it coincides with relative singular cohomology in the following sense: If $(X,A)$ is a relative CW-complex and $G$ is a constant sheaf on $X$, there is a canonical isomorphism $H^i_A(X,G) \cong H^i_{\mathrm{sing}}(X,X \setminus A,G)$.

If you are not satisfied with the proofs for the isomorphism you ask for: What do you mean by explicitly? In which form do you want to represent the elements in both cohomology groups?

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Thanks! What I meant was: for example if I have a element of Hi−1(X−Z,F|X−Z) given by a covering and Cech cycles, how can I see the image in HiZ(X,F)(X)=HiZ(X,F) in terms of the covering. If it is possible how can I understand HiZ(X,F) in terms of Cech Cohomology? – Andrea Jul 14 '12 at 21:45
There is an interpretation of local cohomology via Cech cohomology; see for example Brodmann, M., Sharp, R., Local Cohomology: an Algebraic Introduction with Geometric Applications, Cambridge University Press, 1998. – Martin Brandenburg Jul 15 '12 at 7:44