The Stacks project

Remark 20.29.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a filtered complex of $\mathcal{O}_ X$-modules. If $\mathcal{F}^\bullet $ is bounded from below and for each $n$ the filtration on $\mathcal{F}^ n$ is finite, then there is a construction of the spectral sequence in Lemma 20.29.1 avoiding Injectives, Lemma 19.13.7. Namely, by Derived Categories, Lemma 13.26.9 there is a filtered quasi-isomorphism $i : \mathcal{F}^\bullet \to \mathcal{I}^\bullet $ of filtered complexes with $\mathcal{I}^\bullet $ bounded below, the filtration on $\mathcal{I}^ n$ is finite for all $n$, and with each $\text{gr}^ p\mathcal{I}^ n$ an injective $\mathcal{O}_ X$-module. Then we take the spectral sequence associated to

\[ \Gamma (X, \mathcal{I}^\bullet ) \quad \text{with}\quad F^ p\Gamma (X, \mathcal{I}^\bullet ) = \Gamma (X, F^ p\mathcal{I}^\bullet ) \]

Since cohomology can be computed by evaluating on bounded below complexes of injectives we see that the $E_1$ page is as stated in the lemma. The convergence and boundedness under the stated conditions follows from Homology, Lemma 12.24.11. In fact, this is a special case of the spectral sequence in Derived Categories, Lemma 13.26.14.

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