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83.8 Cohomology on simplicial sites

Let $\mathcal{C}$ be as in Situation 83.3.3. 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 83.3.5. If $\varphi : [m] \to [n]$ is a morphism of $\Delta $, then the diagram of topoi

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \ar[rd]_{g_ n} \ar[rr]_{f_\varphi } & & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ m) \ar[ld]^{g_ m} \\ & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) } \]

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 83.8.1. In Situation 83.3.3 and with notation as above there is a complex

\[ \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z} \]

of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.

Proof. We will use the description of the functors $g_{n!}$ in Lemma 83.3.5 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

\[ \ldots \to \bigoplus \nolimits _{\alpha \in \mathop{Mor}\nolimits _\Delta ([2], [m])]} \mathbf{Z} \to \bigoplus \nolimits _{\alpha \in \mathop{Mor}\nolimits _\Delta ([1], [m])]} \mathbf{Z} \to \bigoplus \nolimits _{\alpha \in \mathop{Mor}\nolimits _\Delta ([0], [m])]} \mathbf{Z} \]

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 83.8.2. In Situation 83.3.3. Let $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}_{total}$ there is a canonical complex

\[ 0 \to \Gamma (\mathcal{C}_{total}, \mathcal{F}) \to \Gamma (\mathcal{C}_0, \mathcal{F}_0) \to \Gamma (\mathcal{C}_1, \mathcal{F}_1) \to \Gamma (\mathcal{C}_2, \mathcal{F}_2) \to \ldots \]

which is exact in degrees $-1, 0$ and exact everywhere if $\mathcal{F}$ is injective.

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 83.8.1 and the fact that $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is exact if $\mathcal{F}$ is injective. $\square$

Lemma 83.8.3. In Situation 83.3.3. For $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral sequence $(E_ r, d_ r)_{r \geq 0}$ with

\[ E_1^{p, q} = H^ q(\mathcal{C}_ p, K_ p),\quad d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q} \]

converging to $H^{p + q}(\mathcal{C}_{total}, K)$. This spectral sequence is functorial in $K$.

Proof. Let $\mathcal{I}^\bullet $ be a bounded below complex of injectives representing $K$. Consider the double complex with terms

\[ A^{p, q} = \Gamma (\mathcal{C}_ p, \mathcal{I}^ q_ p) \]

where the horizontal arrows come from Lemma 83.8.2 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 83.3.5) 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 83.3.6). 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. $\square$

Lemma 83.8.4. Let $\mathcal{C}$ be as in Situation 83.3.3. 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 83.3.1 in case (A) and Lemma 83.3.2 in case (B). The equality then follows from Lemma 83.3.6 and the definition of cohomology. $\square$

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