The Stacks project

Proof. Let ${\mathcal I}^\bullet$ be a bounded below complex of injectives. The map (20.25.0.1) for $\mathcal{I}^\bullet$ is a map $\Gamma (X, {\mathcal I}^\bullet ) \to \text{Tot}(\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet ))$. This is a quasi-isomorphism of complexes of abelian groups as follows from Homology, Lemma 12.25.4 applied to the double complex $\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^\bullet )$ using Lemma 20.11.1. Suppose ${\mathcal F}^\bullet \to {\mathcal I}^\bullet$ is a quasi-isomorphism of ${\mathcal F}^\bullet$ into a bounded below complex of injectives. Since $R\Gamma (X, {\mathcal F}^\bullet )$ is represented by the complex $\Gamma (X, {\mathcal I}^\bullet )$ we obtain the map of the lemma using

We omit the verification of functoriality and compatibilities. To construct the spectral sequence of the lemma, choose a Cartan-Eilenberg resolution $\mathcal{F}^\bullet \to \mathcal{I}^{\bullet , \bullet }$, see Derived Categories, Lemma 13.21.2. In this case $\mathcal{F}^\bullet \to \text{Tot}(\mathcal{I}^{\bullet , \bullet })$ is an injective resolution and hence

computes $R\Gamma (X, \mathcal{F}^\bullet )$ as we've seen above. By Homology, Remark 12.18.4 we can view this as the total complex associated to the triple complex $\check{\mathcal{C}}^\bullet ({\mathcal U}, {\mathcal I}^{\bullet , \bullet })$ hence, using the same remark we can view it as the total complex associate to the double complex $A^{\bullet , \bullet }$ with terms

Since $\mathcal{I}^{q, \bullet }$ is an injective resolution of $\mathcal{F}^ q$ we can apply the first spectral sequence associated to $A^{\bullet , \bullet }$ (Homology, Lemma 12.25.1) to get a spectral sequence with

which is the $n$th term of the complex $\text{Tot}(\check{\mathcal{C}}^\bullet (\mathcal{U}, \underline{H}^ m(\mathcal{F}^\bullet ))$. Hence we obtain $E_2$ terms as described in the lemma. Convergence by Homology, Lemma 12.25.3. $\square$