Lemma 85.9.2. In Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. For an abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ there is a canonical complex

\[ 0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to a_{2, *}\mathcal{F}_2 \to \ldots \]

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

**Proof.**
To construct the complex, by the Yoneda lemma, it suffices for any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$ to construct a complex

\[ 0 \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}, a_*\mathcal{F}) \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}, a_{0, *}\mathcal{F}_0) \to \mathop{\mathrm{Hom}}\nolimits (\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots \]

functorially in $\mathcal{G}$. To do this apply $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ to the exact complex of Lemma 85.9.1 and use adjointness of pullback and pushforward. The exactness properties in degrees $-1, 0$ follow from the construction as $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is left exact. If $\mathcal{F}$ is an injective abelian sheaf, then the complex is exact because $\mathop{\mathrm{Hom}}\nolimits (-, \mathcal{F})$ is exact.
$\square$

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