The Stacks project

Lemma 85.2.9. Let $X$ be a simplicial topological space. The complex of abelian presheaves on $X_{Zar}$

\[ \ldots \to \mathbf{Z}_{X_2} \to \mathbf{Z}_{X_1} \to \mathbf{Z}_{X_0} \]

with boundary $\sum (-1)^ i d^ n_ i$ is a resolution of the constant presheaf $\mathbf{Z}$.

Proof. Let $U \subset X_ m$ be an object of $X_{Zar}$. Then the value of the complex above on $U$ is the complex of abelian groups

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

In other words, this is the complex associated to the free abelian group 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 groups” is a functor, we see that the complex above is homotopy equivalent to the free abelian group 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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