The Stacks project

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

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

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

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. $\square$

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