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 $i>\dim X$.
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7$\begingroup$ Every quasi-coherent sheaf on a quasi-projective scheme is an increasing filtering colimit (i.e., direct limit) of coherent sheaves. Every coherent sheaf "supported" on $Z$ equals the pushforward from the closed subscheme $Z_n$ defined by the $n^\text{th}$ power of the ideal sheaf of $Z$. Pushforward is exact, so preserves cohomology. Cohomology on a Noetherian scheme commutes with filtering direct limits. $\endgroup$– Jason StarrCommented Sep 22, 2018 at 14:36
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$\begingroup$ @JasonStarr Thank you for the answer. $\endgroup$– ChenCommented Sep 24, 2018 at 9:24
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1 Answer
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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.
