Let $(X,\mathcal O_X)$ be a Noetherian Scheme . Let $\mathcal F$ be a coherent sheaf of $\mathcal O_X$-module. Let $Z$ be a closed subscheme of $X$. Let $Y:=Supp \mathcal F$, which is a closed subset of $X$.
Is it true that $H^j_Z(X,\mathcal F)\cong H^j_{Z\cap Y}(Y,\mathcal F|_Y),\forall j\ge 0$ ? (of-course this is true when $X$ is affine ... )
If this is true, then is it also true that $\underline {H^j_Z}(\mathcal F)|_Y\cong \underline {H^j_{Z\cap Y}}(\mathcal F|_Y),\forall j\ge 0$ ?
(Here $\underline {H^j_Z}$ denotes the Sheaf of local Cohomology which is the sheaf associated to the pre-sheaf $U\to H^j_{Z\cap U}(U,\mathcal F|_U)$ , or equivalently it is the right derived functors of the left-exact functor from $Sh(X)$ to $Sh(X)$ which takes a sheaf $\mathcal G$ to the sheaf $U\to \Gamma_{Z\cap U}(U,\mathcal G|_U)$. For reference, see 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 )
Let $X$ be the scheme with coherent sheaf $\mathcal{F}$, let $i:Y \to X$ be the inclusion of the scheme-theoretic support and let $U$ be the open complement of $Z$. Then we know that $\mathcal{F} = i_{\ast} \mathcal{G}$ where $\mathcal{G}$ is a coherent sheaf on $Y$. We have long exact sequences $(A)$ $$ \cdots \to H^i_Z(X, \mathcal{F}) \to H^i(X, \mathcal{F}) \to H^i(U, \mathcal{F}) \to \cdots $$ and $(B)$ $$ \cdots \to H^i_{Y \cap Z}(Y, \mathcal{F}) \to H^i(Y, \mathcal{F}) \to H^i(U \cap Y) \to \cdots $$ together with a map of long exact sequences$ (A) \to (B)$. Moreover we know that $H(X, \mathcal{F})=H(Y, \mathcal{G})$ and $H(U, \mathcal{F})=H(U \cap Y, \mathcal{G})$ and so we conclude that $H^i_Z(X, \mathcal{F})=H^i_{Y \cap Z}(Y, \mathcal{F})$ by the five-lemma.