The Stacks project

Remark 21.14.4. As a consequence of the results above we find that Derived Categories, Lemma 13.22.1 applies to a number of situations. For example, given a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ of ringed topoi we have

\[ R\Gamma (\mathcal{D}, Rf_*\mathcal{F}) = R\Gamma (\mathcal{C}, \mathcal{F}) \]

for any sheaf of $\mathcal{O}_\mathcal {C}$-modules $\mathcal{F}$. Namely, for an injective $\mathcal{O}_\mathcal {X}$-module $\mathcal{I}$ the $\mathcal{O}_\mathcal {D}$-module $f_*\mathcal{I}$ is totally acyclic by Lemma 21.14.1 and a totally acyclic sheaf is acyclic for $\Gamma (\mathcal{D}, -)$ by Lemma 21.14.3.

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