Lemma 85.9.3. In Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. For any $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

\[ E_1^{p, q} = R^ qa_{p, *} K_ p,\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} \]

converging to $R^{p + q}a_*K$. 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

\[ A^{p, q} = a_{p, *}\mathcal{I}^ q_ p \]

where the horizontal arrows come from Lemma 85.9.2 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 $a_*\mathcal{I}^ q$ 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 $R^ qa_{p, *}K_ p$. We conclude by applying Homology, Lemmas 12.25.3 and 12.25.4.
$\square$

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