This section is the analogue of Section 85.8 for sheaves of modules.
In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. In statement of the following lemmas we will let $g_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}), \mathcal{O})$ be the morphism of ringed topoi of Lemma 85.6.1. If $\varphi : [m] \to [n]$ is a morphism of $\Delta $, then the diagram of ringed topoi
is not commutative, but there is a $2$-morphism $g_ n \to g_ m \circ f_\varphi $ coming from the maps $\mathcal{F}(\varphi ) : f_\varphi ^{-1}\mathcal{F}_ m \to \mathcal{F}_ n$. See Sites, Section 7.36.
Lemma 85.10.1. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$. There is a complex of $\mathcal{O}$-modules which forms a resolution of $\mathcal{O}$. Here $g_{n!}$ is as in Lemma 85.6.1.
\[ \ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0 \]
Proof. We will use the description of $g_{n!}$ given in Lemma 85.3.5. As maps of the complex we take $\sum (-1)^ i d^ n_ i$ where $d^ n_ i : g_{n!}\mathcal{O}_ n \to g_{n - 1!}\mathcal{O}_{n - 1}$ is the adjoint to the map $\mathcal{O}_ n \to \bigoplus _{[n - 1] \to [n]} \mathcal{O}_ n = g_ n^*g_{n - 1!}\mathcal{O}_{n - 1}$ corresponding to the factor labeled with $\delta ^ n_ i : [n - 1] \to [n]$. Then $g_ m^{-1}$ applied to the complex gives the complex
on $\mathcal{C}_ m$. In other words, this is the complex associated to the free $\mathcal{O}_ m$-module on the simplicial set $\Delta [m]$, see Simplicial, Example 14.11.2. Since $\Delta [m]$ is homotopy equivalent to $\Delta [0]$, see Simplicial, Example 14.26.7, and since “taking free abelian sheaf on” is a functor, we see that the complex above is homotopy equivalent to the free abelian sheaf on $\Delta [0]$ (Simplicial, Remark 14.26.4 and Lemma 14.27.2). This complex is acyclic in positive degrees and equal to $\mathcal{O}_ m$ in degree $0$. $\square$
Lemma 85.10.2. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules. There is a canonical complex which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is an injective $\mathcal{O}$-module.
\[ 0 \to \Gamma (\mathcal{C}_{total}, \mathcal{F}) \to \Gamma (\mathcal{C}_0, \mathcal{F}_0) \to \Gamma (\mathcal{C}_1, \mathcal{F}_1) \to \Gamma (\mathcal{C}_2, \mathcal{F}_2) \to \ldots \]
Proof. Observe that $\mathop{\mathrm{Hom}}\nolimits (\mathcal{O}, \mathcal{F}) = \Gamma (\mathcal{C}_{total}, \mathcal{F})$ and $\mathop{\mathrm{Hom}}\nolimits (g_{n!}\mathcal{O}_ n, \mathcal{F}) = \Gamma (\mathcal{C}_ n, \mathcal{F}_ n)$. Hence this lemma is an immediate consequence of Lemma 85.10.1 and the fact that $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is exact if $\mathcal{F}$ is injective. $\square$
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 converging to $H^{p + q}(\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$.
\[ E_1^{p, q} = H^ q(\mathcal{C}_ p, K_ p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} \]
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 85.10.2 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. $\square$
Lemma 85.10.4. In Situation 85.3.3 let $\mathcal{O}$ be a sheaf of rings. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$. Let $\mathcal{F} \in \textit{Mod}(\mathcal{O})$. Then $H^ p(U, \mathcal{F}) = H^ p(U, g_ n^*\mathcal{F})$ where on the left hand side $U$ is viewed as an object of $\mathcal{C}_{total}$.
Proof. Observe that “$U$ viewed as object of $\mathcal{C}_{total}$” is explained by the construction of $\mathcal{C}_{total}$ in Lemma 85.3.1 in case (A) and Lemma 85.3.2 in case (B). In both cases the functor $\mathcal{C}_ n \to \mathcal{C}$ is continuous and cocontinuous, see Lemma 85.3.5, and $g_ n^{-1}\mathcal{O} = \mathcal{O}_ n$ by definition. Hence the result is a special case of Cohomology on Sites, Lemma 21.37.5. $\square$
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