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

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

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).