The Stacks project

Lemma 85.9.1. In Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. For any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$ there is an exact complex

\[ \ldots \to g_{2!}(a_2^{-1}\mathcal{G}) \to g_{1!}(a_1^{-1}\mathcal{G}) \to g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G} \to 0 \]

of abelian sheaves on $\mathcal{C}_{total}$.

Proof. We encourage the reader to read the proof of Lemma 85.8.1 first. We will use Lemma 85.4.2 and the description of the functors $g_{n!}$ in Lemma 85.3.5 without further mention. In particular $g_{n!}(a_ n^{-1}\mathcal{G})$ is the sheaf on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ m$ is the sheaf

\[ \bigoplus \nolimits _{\varphi : [n] \to [m]} f_\varphi ^{-1}a_ n^{-1}\mathcal{G} = \bigoplus \nolimits _{\varphi : [n] \to [m]} a_ m^{-1}\mathcal{G} \]

As maps of the complex we take $\sum (-1)^ i d^ n_ i$ where $d^ n_ i : g_{n!}(a_ n^{-1}\mathcal{G}) \to g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$ is the adjoint to the map $a_ n^{-1}\mathcal{G} \to \bigoplus _{[n - 1] \to [n]} a_ n^{-1}\mathcal{G} = g_ n^{-1}g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$ corresponding to the factor labeled with $\delta ^ n_ i : [n - 1] \to [n]$. The map $g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G}$ is adjoint to the identity map of $a_0^{-1}\mathcal{G}$. Then $g_ m^{-1}$ applied to the chain complex in degrees $\ldots , 2, 1, 0$ gives the complex

\[ \ldots \to \bigoplus \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits _\Delta ([2], [m])]} a_ m^{-1}\mathcal{G} \to \bigoplus \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits _\Delta ([1], [m])]} a_ m^{-1}\mathcal{G} \to \bigoplus \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits _\Delta ([0], [m])]} a_ m^{-1}\mathcal{G} \]

on $\mathcal{C}_ m$. This is equal to $a_ m^{-1}\mathcal{G}$ tensored over the constant sheaf $\mathbf{Z}$ with the complex

\[ \ldots \to \bigoplus \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits _\Delta ([2], [m])]} \mathbf{Z} \to \bigoplus \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits _\Delta ([1], [m])]} \mathbf{Z} \to \bigoplus \nolimits _{\alpha \in \mathop{\mathrm{Mor}}\nolimits _\Delta ([0], [m])]} \mathbf{Z} \]

discussed in the proof of Lemma 85.8.1. There we have seen that this complex is homotopy equivalent to $\mathbf{Z}$ placed in degree $0$ which finishes the proof. $\square$

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