Let $\mathcal{C}$ be as in Situation 85.3.3. In statement of the following lemmas we will let $g_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ be the morphism of topoi of Lemma 85.3.5. If $\varphi : [m] \to [n]$ is a morphism of $\Delta $, then the diagram of 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 85.8.1. In Situation 85.3.3 and with notation as above there is a complex of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.
\[ \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z} \]
Proof. We will use the description of the functors $g_{n!}$ in Lemma 85.3.5 without further mention. As maps of the complex we take $\sum (-1)^ i d^ n_ i$ where $d^ n_ i : g_{n!}\mathbf{Z} \to g_{n - 1!}\mathbf{Z}$ is the adjoint to the map $\mathbf{Z} \to \bigoplus _{[n - 1] \to [n]} \mathbf{Z} = g_ n^{-1}g_{n - 1!}\mathbf{Z}$ 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 abelian sheaf 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 $\mathbf{Z}$ in degree $0$. $\square$
Lemma 85.8.2. In Situation 85.3.3. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{total}$ there is a canonical complex which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is injective.
\[ 0 \to \Gamma (\mathcal{C}_{total}, \mathcal{F}) \to \Gamma (\mathcal{C}_0, \mathcal{F}_0) \to \Gamma (\mathcal{C}_1, \mathcal{F}_1) \to \Gamma (\mathcal{C}_2, \mathcal{F}_2) \to \ldots \]
Proof. Observe that $\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, \mathcal{F}) = \Gamma (\mathcal{C}_{total}, \mathcal{F})$ and $\mathop{\mathrm{Hom}}\nolimits (g_{n!}\mathbf{Z}, \mathcal{F}) = \Gamma (\mathcal{C}_ n, \mathcal{F}_ n)$. Hence this lemma is an immediate consequence of Lemma 85.8.1 and the fact that $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is exact if $\mathcal{F}$ is injective. $\square$
Lemma 85.8.3. In Situation 85.3.3. For $K$ in $D^+(\mathcal{C}_{total})$ 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 injectives representing $K$. Consider the double complex with terms
where the horizontal arrows come from Lemma 85.8.2 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 $\Gamma (\mathcal{C}_{total}, \mathcal{I}^ q)$ 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 the groups $H^ q(\mathcal{C}_ p, K_ p)$. We conclude by applying Homology, Lemmas 12.25.3 and 12.25.4. $\square$
Remark 85.8.4. Assumptions and notation as in Lemma 85.8.3 except we do not require $K$ in $D(\mathcal{C}_{total})$ to be bounded below. We claim there is a natural spectral sequence in this case also. Namely, suppose that $\mathcal{I}^\bullet $ is a K-injective complex of sheaves on $\mathcal{C}_{total}$ with injective terms representing $K$. We have where $A^{\bullet , \bullet }$ is the double complex with terms $A^{p, q} = \Gamma (\mathcal{C}_ p, \mathcal{I}^ q_ p)$ and $\text{Tot}_\pi $ denotes the product totalization of this double complex. Namely, the first equality holds in any site. The second equality holds by Lemma 85.8.1. The third equality holds because $\mathcal{I}^\bullet $ is K-injective, see Cohomology on Sites, Sections 21.34 and 21.35. The final equality holds by the construction of $\mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet $ and the fact that $\mathop{\mathrm{Hom}}\nolimits (g_{p!}\mathbf{Z}, \mathcal{I}^ q) = \Gamma (\mathcal{C}_ p, \mathcal{I}^ q_ p)$. Then we get our spectral sequence by viewing $\text{Tot}_\pi (A^{\bullet , \bullet })$ as a filtered complex with $F^ i\text{Tot}^ n_\pi (A^{\bullet , \bullet }) = \prod _{p + q = n,\ p \geq i} A^{p, q}$. The spectral sequence we obtain behaves like the spectral sequence $({}'E_ r, {}'d_ r)_{r \geq 0}$ in Homology, Section 12.25 (where the case of the direct sum totalization is discussed) except for regularity, boundedness, convergence, and abutment issues. In particular we obtain $E_1^{p, q} = H^ q(\mathcal{C}_ p, K_ p)$ as in Lemma 85.8.3.
\begin{align*} R\Gamma (\mathcal{C}_{total}, K) & = R\mathop{\mathrm{Hom}}\nolimits (\mathbf{Z}, K) \\ & = R\mathop{\mathrm{Hom}}\nolimits ( \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, K) \\ & = \Gamma (\mathcal{C}_{total}, \mathop{\mathcal{H}\! \mathit{om}}\nolimits ^\bullet ( \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}, \mathcal{I}^\bullet )) \\ & = \text{Tot}_\pi (A^{\bullet , \bullet }) \end{align*}
Lemma 85.8.5. In Situation 85.3.3. Let $K$ be an object of $D(\mathcal{C}_{total})$.
If $H^{-p}(\mathcal{C}_ p, K_ p) = 0$ for all $p \geq 0$, then $H^0(\mathcal{C}_{total}, K) = 0$.
If $R\Gamma (\mathcal{C}_ p, K_ p) = 0$ for all $p \geq 0$, then $R\Gamma (\mathcal{C}_{total}, K) = 0$.
Proof. With notation as in Remark 85.8.4 we see that $R\Gamma (\mathcal{C}_{total}, K)$ is represented by $\text{Tot}_\pi (A^{\bullet , \bullet })$. The assumption in (1) tells us that $H^{-p}(A^{p, \bullet }) = 0$. Thus the vanishing in (1) follows from More on Algebra, Lemma 15.103.1. Part (2) follows from part (1) and taking shifts. $\square$
Lemma 85.8.6. Let $\mathcal{C}$ be as in Situation 85.3.3. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}_ n)$. Let $\mathcal{F} \in \textit{Ab}(\mathcal{C}_{total})$. Then $H^ p(U, \mathcal{F}) = H^ p(U, g_ n^{-1}\mathcal{F})$ 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). The equality then follows from Lemma 85.3.6 and the definition of cohomology. $\square$
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