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

$E_1^{p, q} = R^{p + q}f_*\text{gr}^ p\mathcal{F}^\bullet$

If for every $n$ we have

$R^ nf_*F^ p\mathcal{F}^\bullet = 0 \text{ for }p \gg 0 \quad \text{and}\quad R^ nf_*F^ p\mathcal{F}^\bullet = R^ nf_*\mathcal{F}^\bullet \text{ for }p \ll 0$

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

Proof. The proof is exactly the same as the proof of Lemma 20.29.1. $\square$

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