The Stacks project

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

\[ E_2^{p, q} = H^ p(\mathcal{D}, R^ qf_*(\mathcal{F}^\bullet )) \]

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

Proof. This is just the Grothendieck spectral sequence Derived Categories, Lemma 13.22.2 coming from the composition of functors $\Gamma (\mathcal{C}, -) = \Gamma (\mathcal{D}, -) \circ f_*$. To see that the assumptions of Derived Categories, Lemma 13.22.2 are satisfied, see Lemmas 21.15.1 and 21.15.3. $\square$


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