Lemma 85.11.4. With notation as above for any $K$ in $D^+(\mathcal{O})$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ in $\textit{Mod}(\mathcal{O}_\mathcal {D})$ with
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 injective $\mathcal{O}$-modules representing $K$. Consider the double complex with terms
where the horizontal arrows come from Lemma 85.11.3 and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet $. The lemma says rows of the double complex are exact in positive degrees and evaluate to $a_*\mathcal{I}^ q$ in degree $0$. Thus the total complex associated to the double complex computes $Ra_*K$ by Homology, Lemma 12.25.4. 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 totally acyclic on $\mathcal{C}_ p$ (Lemma 85.6.2). Hence the cohomology of the columns are the sheaves $R^ qa_{p, *}K_ p$ by Leray's acyclicity lemma (Derived Categories, Lemma 13.16.7) and Cohomology on Sites, Lemma 21.14.3. We conclude by applying Homology, Lemma 12.25.3. $\square$
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