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.
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.
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. $\square$
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