Let $X$ be a smooth, projective variety, ${F}$ a quasi-coherent $\mathcal{O}\_X$-module on $X$ supported on a closed subscheme, say $Z \subset X$. Is it true that $H^i(X,F)=0$ for all $i>\dim Z$? 


We know that $H^i(X,F)=0$ for all $i>\dim X$.



Reformulation of Grothendieck vanishing theorem
Every abelian sheaf $\mathcal{F}$ on $X$ whose support is contained in $Z$, has vanishing cohomology groups $H^i(X, \mathcal{F})$ for $i > \dim Z$.


Proof. The support of an abelian sheaf is the set of points where the stalk is nonzero. If the support of $\mathcal{F}$ is contained in $Z$, then $\mathcal{F}$ is equal to $i\_\*(i^{-1}\mathcal{F})$ where
$i : Z \to X$ is the inclusion map (look at stalks). Since $i\_\*$ is exact, we have $H^a(X, i\_\*\mathcal{G}) = H^a(Z, \mathcal{G})$ for any abelian sheaf $\mathcal{G}$ on $Z$ and every $a$ (for example by the Leray spectral sequence, although this is overkill). Thus Grothendieck's vanishing for sheaves on $Z$ gives the result you are looking for.


Warning: when you apply this make sure your notion of support agrees with the notion given above!


PS: this is the same as Jason's answer above.


