The Stacks project

Lemma 84.8.1. In Situation 84.3.3 and with notation as above there is a complex

\[ \ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z} \]

of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.

Proof. We will use the description of the functors $g_{n!}$ in Lemma 84.3.5 without further mention. As maps of the complex we take $\sum (-1)^ i d^ n_ i$ where $d^ n_ i : g_{n!}\mathbf{Z} \to g_{n - 1!}\mathbf{Z}$ is the adjoint to the map $\mathbf{Z} \to \bigoplus _{[n - 1] \to [n]} \mathbf{Z} = g_ n^{-1}g_{n - 1!}\mathbf{Z}$ corresponding to the factor labeled with $\delta ^ n_ i : [n - 1] \to [n]$. Then $g_ m^{-1}$ applied to the complex gives the complex

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

on $\mathcal{C}_ m$. In other words, this is the complex associated to the free abelian sheaf on the simplicial set $\Delta [m]$, see Simplicial, Example 14.11.2. Since $\Delta [m]$ is homotopy equivalent to $\Delta [0]$, see Simplicial, Example 14.26.7, and since “taking free abelian sheaf on” is a functor, we see that the complex above is homotopy equivalent to the free abelian sheaf on $\Delta [0]$ (Simplicial, Remark 14.26.4 and Lemma 14.27.2). This complex is acyclic in positive degrees and equal to $\mathbf{Z}$ in degree $0$. $\square$

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