Lemma 21.14.5 (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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