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 $0$. $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 85.2: Simplicial topological spaces

Post a comment

Your email address will not be published. Required fields are marked.

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09W5. Beware of the difference between the letter 'O' and the digit '0'.