Lemma 84.2.9. 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 $0$. $\square$

There are also:

• 2 comment(s) on Section 84.2: Simplicial topological spaces

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).