Lemma 85.8.2 (0D77)—The Stacks project

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Table of contents
Part 5: Topics in Geometry
Chapter 85: Simplicial Spaces
Section 85.8: Cohomology on simplicial sites
Lemma 85.8.2 (cite)

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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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