The Stacks project

Lemma 85.10.3. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings. For $K$ in $D^+(\mathcal{O})$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

\[ E_1^{p, q} = H^ q(\mathcal{C}_ p, K_ p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} \]

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 injective $\mathcal{O}$-modules representing $K$. Consider the double complex with terms

\[ A^{p, q} = \Gamma (\mathcal{C}_ p, \mathcal{I}^ q_ p) \]

where the horizontal arrows come from Lemma 85.10.2 and the vertical arrows from the differentials of the complex $\mathcal{I}^\bullet $. Observe that $\Gamma (\mathcal{D}, -) = \mathop{\mathrm{Hom}}\nolimits _{\mathcal{O}_\mathcal {D}}(\mathcal{O}_\mathcal {D}, -)$ on $\textit{Mod}(\mathcal{O}_\mathcal {D})$. Hence the lemma says rows of the double complex are exact in positive degrees and evaluate to $\Gamma (\mathcal{C}_{total}, \mathcal{I}^ q)$ in degree $0$. Thus the total complex associated to the double complex computes $R\Gamma (\mathcal{C}_{total}, K)$ 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 computes the groups $H^ q(\mathcal{C}_ p, K_ p)$ 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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