Lemma 21.15.5 (Leray spectral sequence). Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be a morphism of ringed topoi. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_\mathcal {C}$-modules. There is a spectral sequence

converging to $H^{p + q}(\mathcal{C}, \mathcal{F}^\bullet )$.

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