The Stacks project

Lemma 20.13.4 (Leray spectral sequence). Let $f : X \to Y$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet $ be a bounded below complex of $\mathcal{O}_ X$-modules. There is a spectral sequence

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

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

Proof. This is just the Grothendieck spectral sequence Derived Categories, Lemma 13.22.2 coming from the composition of functors $\Gamma _{res} = \Gamma (Y, -) \circ f_*$ where $\Gamma _{res}$ is as in the proof of Lemma 20.13.1. To see that the assumptions of Derived Categories, Lemma 13.22.2 are satisfied, see the proof of Lemma 20.13.1 or Remark 20.13.2. $\square$

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