The Stacks project

Example 20.29.3. 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 = \tau _{\leq -p}\mathcal{F}^\bullet $. (If $\mathcal{F}^\bullet $ is bounded below we can use Remark 20.29.2.) Then we get a spectral sequence

\[ E_1^{p, q} = H^{p + q}(X, H^{-p}(\mathcal{F}^\bullet )[p]) = H^{2p + q}(X, H^{-p}(\mathcal{F}^\bullet )) \]

After renumbering $p = -j$ and $q = i + 2j$ we find that for any $K \in D(\mathcal{O}_ X)$ there is a spectral sequence $(E'_ r, d'_ r)_{r \geq 2}$ of bigraded modules with $d'_ r$ of bidegree $(r, -r + 1)$, with

\[ (E'_2)^{i, j} = H^ i(X, H^ j(K)) \]

If $K$ is bounded below (for example), then this spectral sequence is bounded and converges to $H^{i + j}(X, K)$. In the bounded below case this spectral sequence is an example of the second spectral sequence of Derived Categories, Lemma 13.21.3 (constructed using Cartan-Eilenberg resolutions).

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