Local cohomology as direct limit of Ext functors, for not necessarily affine schemes?

Question

Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , where $i:Z\to X$ is the inclusion and $\mathcal J$ is the kernel of $i^{\#}: \mathcal O_X \to i_*(\mathcal O_Z) $.

Let $\mathcal F$ be a quasi-coherent sheaf of $\mathcal O_X$-modules on $X$.

Now if $X$ is affine, then it follows that $\varinjlim_{n} \mathcal Ext^j(\mathcal O_X/\mathcal J^n,\mathcal F)\cong \underline {H^j_Z} (\mathcal F), \forall j\ge 0$.

My question is: Is there any known general cases where the above isomorphism or some analogue of it holds when $X$ is not necessarily affine ?

Answer 1

What I want is given as Theorem 2.8, R. Hartshorne. Local Cohomology. A seminar given by A. Grothendieck, Harvard University. Fall, 1961. Lecture Notes in Mathematics, 41. Springer, 1967. http://link.springer.com/book/10.1007%2FBFb0073971