Lemma 21.8.4. Let $f : (\mathcal{C}, \mathcal{O}_\mathcal {C}) \to (\mathcal{D}, \mathcal{O}_\mathcal {D})$ be a morphism of ringed sites corresponding to the continuous functor $u : \mathcal{D} \to \mathcal{C}$. For any $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\textit{Mod}(\mathcal{O}_\mathcal {C}))$ the sheaf $R^ if_*\mathcal{F}$ is the sheaf associated to the presheaf

**Proof.**
Let $\mathcal{F} \to \mathcal{I}^\bullet $ be an injective resolution. Then $R^ if_*\mathcal{F}$ is by definition the $i$th cohomology sheaf of the complex

By definition of the abelian category structure on $\mathcal{O}_\mathcal {D}$-modules this cohomology sheaf is the sheaf associated to the presheaf

and this is obviously equal to

which is equal to $H^ i(u(V), \mathcal{F})$ and we win. $\square$

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## Comments (2)

Comment #2169 by Kestutis Cesnavicius on

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