Lemma 85.8.3. In Situation 85.3.3. For $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with
converging to $H^{p + q}(\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$.
Proof. Let $\mathcal{I}^\bullet $ be a bounded below complex of injectives representing $K$. Consider the double complex with terms
where the horizontal arrows come from Lemma 85.8.2 and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet $. The rows of the double complex are exact in positive degrees and evaluate to $\Gamma (\mathcal{C}_{total}, \mathcal{I}^ q)$ in degree $0$. On the other hand, since restriction to $\mathcal{C}_ p$ is exact (Lemma 85.3.5) the complex $\mathcal{I}_ p^\bullet $ represents $K_ p$ in $D(\mathcal{C}_ p)$. The sheaves $\mathcal{I}_ p^ q$ are injective abelian sheaves on $\mathcal{C}_ p$ (Lemma 85.3.6). Hence the cohomology of the columns computes the groups $H^ q(\mathcal{C}_ p, K_ p)$. We conclude by applying Homology, Lemmas 12.25.3 and 12.25.4. $\square$
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