Lemma 21.14.7 (Relative 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}) and g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{E}), \mathcal{O}_\mathcal {E}) be morphisms of ringed topoi. Let \mathcal{F} be an \mathcal{O}_\mathcal {C}-module. There is a spectral sequence with
converging to R^{p + q}(g \circ f)_*\mathcal{F}. This spectral sequence is functorial in \mathcal{F}, and there is a version for bounded below complexes of \mathcal{O}_\mathcal {C}-modules.
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