Lemma 20.29.5. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces. Let $\mathcal{F}^\bullet $ be a filtered complex of $\mathcal{O}_ X$-modules. There exists a canonical spectral sequence $(E_ r, \text{d}_ r)_{r \geq 1}$ of bigraded $\mathcal{O}_ Y$-modules with $d_ r$ of bidegree $(r, -r + 1)$ and

If for every $n$ we have

then the spectral sequence is bounded and converges to $Rf_*\mathcal{F}^\bullet $.

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