The Stacks project

Lemma 20.13.8 (Relative Leray spectral sequence). Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of ringed spaces. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. There is a spectral sequence with

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

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 $\mathcal{O}_ X$-modules.

Proof. This is a Grothendieck spectral sequence for composition of functors and follows from Lemma 20.13.7 and Derived Categories, Lemma 13.22.2. $\square$

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