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The Stacks project

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

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 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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