Let $(X,\mathcal O_X)$ be a Noetherian, affine Scheme and $\mathcal F$ be a quasi-coherent Sheaf of $\mathcal O_X$-modules on $X$. Let $\dim \mathcal F$ be the Krull dimension of $\{x\in X| \mathcal F_x\ne0\}$. Let $Z$ be a closed subset of $X$, then it is known that the local cohomology $H^i_Z(X,\mathcal F)=0,\forall i>\dim \mathcal F$ .
My question is: Is any such result, for vanishing of Local Cohomology, known for general, not necessarily affine, Noetherian schemes (even for coherent sheaves) or at least say for projective schemes ?