Lemma 52.6.20. Let $A$ be a ring and let $I \subset A$ be a finitely generated ideal. Let $\mathcal{C}$ be a site and let $\mathcal{O}$ be a sheaf of $A$-algebras. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. Then we have

$R\Gamma (\mathcal{C}, \mathcal{F})^\wedge = R\Gamma (\mathcal{C}, \mathcal{F}^\wedge )$

in $D(A)$ where $\mathcal{F}^\wedge$ is the derived completion of $\mathcal{F}$ with respect to $I\mathcal{O}$ and on the left hand wide we have the derived completion with respect to $I$. This produces two spectral sequences

$E_2^{i, j} = H^ i(H^ j(\mathcal{C}, \mathcal{F})^\wedge ) \quad \text{and}\quad E_2^{p, q} = H^ p(\mathcal{C}, H^ q(\mathcal{F}^\wedge ))$

both converging to $H^*(R\Gamma (\mathcal{C}, \mathcal{F})^\wedge ) = H^*(\mathcal{C}, \mathcal{F}^\wedge )$

Proof. Apply Lemma 52.6.19 to the morphism of ringed topoi $(\mathcal{C}, \mathcal{O}) \to (pt, A)$ and take cohomology to get the first statement. The second spectral sequence is the second spectral sequence of Derived Categories, Lemma 13.21.3. The first spectral sequence is the spectral sequence of More on Algebra, Example 15.91.22 applied to $R\Gamma (\mathcal{C}, \mathcal{F})^\wedge$. $\square$

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