The Stacks project

Example 20.29.4. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}^\bullet $ be a complex of $\mathcal{O}_ X$-modules. We can apply Lemma 20.29.1 with $F^ p\mathcal{F}^\bullet = \sigma _{\geq p}\mathcal{F}^\bullet $. Then we get a spectral sequence

\[ E_1^{p, q} = H^{p + q}(X, \mathcal{F}^ p[-p]) = H^ q(X, \mathcal{F}^ p) \]

If $\mathcal{F}^\bullet $ is bounded below, then

  1. we can use Remark 20.29.2 to construct this spectral sequence,

  2. the spectral sequence is bounded and converges to $H^{i + j}(X, \mathcal{F}^\bullet )$, and

  3. the spectral sequence is equal to the first spectral sequence of Derived Categories, Lemma 13.21.3 (constructed using Cartan-Eilenberg resolutions).

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