Consider a simplicial site $\mathcal{C}$ as in Situation 85.3.3. Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. By Lemma 85.4.2 we obtain a morphism of topoi
and morphisms of topoi $g_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ as in Lemma 85.3.5. The compositions $a \circ g_ n$ are denoted $a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$. Furthermore, the simplicial structure gives morphisms of topoi $f_\varphi : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m)$ such that $a_ n \circ f_\varphi = a_ m$ for all $\varphi : [m] \to [n]$.
Lemma 85.9.1. In Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. For any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$ there is an exact complex of abelian sheaves on $\mathcal{C}_{total}$.
\[ \ldots \to g_{2!}(a_2^{-1}\mathcal{G}) \to g_{1!}(a_1^{-1}\mathcal{G}) \to g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G} \to 0 \]
Proof. We encourage the reader to read the proof of Lemma 85.8.1 first. We will use Lemma 85.4.2 and the description of the functors $g_{n!}$ in Lemma 85.3.5 without further mention. In particular $g_{n!}(a_ n^{-1}\mathcal{G})$ is the sheaf on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the sheaf
As maps of the complex we take $\sum (-1)^ i d^ n_ i$ where $d^ n_ i : g_{n!}(a_ n^{-1}\mathcal{G}) \to g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$ is the adjoint to the map $a_ n^{-1}\mathcal{G} \to \bigoplus _{[n - 1] \to [n]} a_ n^{-1}\mathcal{G} = g_ n^{-1}g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$ corresponding to the factor labeled with $\delta ^ n_ i : [n - 1] \to [n]$. The map $g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G}$ is adjoint to the identity map of $a_0^{-1}\mathcal{G}$. Then $g_ m^{-1}$ applied to the chain complex in degrees $\ldots , 2, 1, 0$ gives the complex
on $\mathcal{C}_ m$. This is equal to $a_ m^{-1}\mathcal{G}$ tensored over the constant sheaf $\mathbf{Z}$ with the complex
discussed in the proof of Lemma 85.8.1. There we have seen that this complex is homotopy equivalent to $\mathbf{Z}$ placed in degree $0$ which finishes the proof. $\square$
Lemma 85.9.2. In Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. For an abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ there is a canonical complex on $\mathcal{D}$ which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is injective.
\[ 0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots \]
Proof. To construct the complex, by the Yoneda lemma, it suffices for any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$ to construct a complex
functorially in $\mathcal{G}$. To do this apply $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ to the exact complex of Lemma 85.9.1 and use adjointness of pullback and pushforward. The exactness properties in degrees $-1, 0$ follow from the construction as $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is left exact. If $\mathcal{F}$ is an injective abelian sheaf, then the complex is exact because $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is exact. $\square$
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 converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.
\[ E_1^{p, q} = R^ qa_{p, *} K_ p,\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} \]
Proof. Let $\mathcal{I}^\bullet $ be a bounded below complex of injectives representing $K$. Consider the double complex with terms
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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