Lemma 21.14.3. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ be a ringed topos. A totally acyclic sheaf is right acyclic for the following functors:

the functor $H^0(U, -)$ for any object $U$ of $\mathcal{C}$,

the functor $\mathcal{F} \mapsto \mathcal{F}(K)$ for any presheaf of sets $K$,

the functor $\Gamma (\mathcal{C}, -)$ of global sections,

the functor $f_*$ for any 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.

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